Litigation (Details) - USD ($) $ in Millions	3 Months Ended
Sep. 30, 2021	Dec. 31, 2023
Settled Lawsuit
Loss Contingencies [Line Items]
Litigation settlement amount	$ 1.1
Insurance coverage receivable		$ 0.9
Minimum \| Loss of Funds by Customer
Loss Contingencies [Line Items]
Lawsuit alleging fault for a loss of funds with possible loss		0.0
Maximum \| Loss of Funds by Customer
Loss Contingencies [Line Items]
Lawsuit alleging fault for a loss of funds with possible loss		$ 1.5

Amount awarded to other party in judgment or settlement of litigation.

No definition available.

Name:	us-gaap_LitigationSettlementAmountAwardedToOtherParty
Namespace Prefix:	us-gaap_
Data Type:	xbrli:monetaryItemType
Balance Type:	credit
Period Type:	duration

- Definition

Line items represent financial concepts included in a table. These concepts are used to disclose reportable information associated with domain members defined in one or many axes to the table.

+ References

Reference 1: http://www.xbrl.org/2003/role/disclosureRef
-Topic 460
-SubTopic 10
-Name Accounting Standards Codification
-Section 50
-Paragraph 3
-Publisher FASB
-URI https://asc.fasb.org//1943274/2147482425/460-10-50-3

Reference 2: http://www.xbrl.org/2003/role/disclosureRef
-Topic 450
-SubTopic 20
-Name Accounting Standards Codification
-Section 50
-Paragraph 1
-Publisher FASB
-URI https://asc.fasb.org//1943274/2147483076/450-20-50-1

Reference 3: http://www.xbrl.org/2009/role/commonPracticeRef
-Topic 450
-SubTopic 20
-Name Accounting Standards Codification
-Section 50
-Paragraph 4
-Subparagraph (a)
-Publisher FASB
-URI https://asc.fasb.org//1943274/2147483076/450-20-50-4

Reference 4: http://www.xbrl.org/2003/role/disclosureRef
-Topic 450
-SubTopic 20
-Name Accounting Standards Codification
-Section 50
-Paragraph 4
-Subparagraph (b)
-Publisher FASB
-URI https://asc.fasb.org//1943274/2147483076/450-20-50-4

Reference 5: http://www.xbrl.org/2009/role/commonPracticeRef
-Topic 450
-SubTopic 20
-Name Accounting Standards Codification
-Section 50
-Paragraph 9
-Subparagraph (a)
-Publisher FASB
-URI https://asc.fasb.org//1943274/2147483076/450-20-50-9

Reference 6: http://www.xbrl.org/2003/role/disclosureRef
-Topic 450
-SubTopic 20
-Name Accounting Standards Codification
-Section 50
-Paragraph 9
-Subparagraph (b)
-Publisher FASB
-URI https://asc.fasb.org//1943274/2147483076/450-20-50-9

+ Details

Name:	us-gaap_LossContingenciesLineItems
Namespace Prefix:	us-gaap_
Data Type:	xbrli:stringItemType
Balance Type:	na
Period Type:	duration

- Definition

Reflects the estimated amount of loss from the specified contingency as of the balance sheet date.

+ References

+ Details

Name:	us-gaap_LossContingencyEstimateOfPossibleLoss
Namespace Prefix:	us-gaap_
Data Type:	xbrli:monetaryItemType
Balance Type:	debit
Period Type:	instant

- Definition

Amount of receivable related to a loss contingency accrual that is expected to be collected within one year or the normal operating cycle, if longer. For example, an insurance recovery receivable.

+ References

Reference 1: http://www.xbrl.org/2009/role/commonPracticeRef
-Topic 460
-SubTopic 10
-Name Accounting Standards Codification
-Section 50
-Paragraph 3
-Publisher FASB
-URI https://asc.fasb.org//1943274/2147482425/460-10-50-3

+ Details

Name:	us-gaap_LossContingencyReceivableCurrent
Namespace Prefix:	us-gaap_
Data Type:	xbrli:monetaryItemType
Balance Type:	debit
Period Type:	instant

- Details

Name:	us-gaap_LossContingenciesByNatureOfContingencyAxis=fgbi_SettledLawsuitMember
Namespace Prefix:
Data Type:	na
Balance Type:
Period Type:

- Details

Name:	srt_RangeAxis=srt_MinimumMember
Namespace Prefix:
Data Type:	na
Balance Type:
Period Type:

- Details

Name:	us-gaap_LossContingenciesByNatureOfContingencyAxis=fgbi_LossOfFundsByCustomerMember
Namespace Prefix:
Data Type:	na
Balance Type:
Period Type:

- Details

Name:	srt_RangeAxis=srt_MaximumMember
Namespace Prefix:
Data Type:	na
Balance Type:
Period Type: