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G = C13⋊S4  order 312 = 23·3·13

The semidirect product of C13 and S4 acting via S4/A4=C2

non-abelian, soluble, monomial

Aliases: C13⋊S4, A4⋊D13, C22⋊D39, (C2×C26)⋊2S3, (A4×C13)⋊1C2, SmallGroup(312,48)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C22 — A4×C13 — C13⋊S4
Chief series
C1 — C22 — C2×C26 — A4×C13 — C13⋊S4
Lower central
A4×C13 — C13⋊S4
Upper central
C1

Generators and relations for C13⋊S4
 G = < a,b,c,d,e | a13=b2=c2=d3=e2=1, ab=ba, ac=ca, ad=da, eae=a-1, dbd-1=ebe=bc=cb, dcd-1=b, ce=ec, ede=d-1 >

C1
3C2
78C2
4C3
C13
C22
39C22
39C4
52S3
3C26
6D13
4C39
39D4
A4
C2×C26
3D26
3Dic13
4D39
13S4
3C13⋊D4
A4×C13
C13⋊S4

Character table of C13⋊S4

 class 12A2B3413A13B13C13D13E13F26A26B26C26D26E26F39A39B39C39D39E39F39G39H39I39J39K39L
 size 1378878222222666666888888888888
ρ111111111111111111111111111111    trivial
ρ211-11-1111111111111111111111111    linear of order 2
ρ3220-10222222222222-1-1-1-1-1-1-1-1-1-1-1-1    orthogonal lifted from S3
ρ422020ζ1311+ζ132ζ139+ζ134ζ138+ζ135ζ137+ζ136ζ1312+ζ13ζ1310+ζ133ζ138+ζ135ζ1311+ζ132ζ137+ζ136ζ139+ζ134ζ1312+ζ13ζ1310+ζ133ζ1310+ζ133ζ139+ζ134ζ138+ζ135ζ137+ζ136ζ137+ζ136ζ138+ζ135ζ139+ζ134ζ1310+ζ133ζ1311+ζ132ζ1312+ζ13ζ1312+ζ13ζ1311+ζ132    orthogonal lifted from D13
ρ522020ζ138+ζ135ζ1310+ζ133ζ137+ζ136ζ1311+ζ132ζ139+ζ134ζ1312+ζ13ζ137+ζ136ζ138+ζ135ζ1311+ζ132ζ1310+ζ133ζ139+ζ134ζ1312+ζ13ζ1312+ζ13ζ1310+ζ133ζ137+ζ136ζ1311+ζ132ζ1311+ζ132ζ137+ζ136ζ1310+ζ133ζ1312+ζ13ζ138+ζ135ζ139+ζ134ζ139+ζ134ζ138+ζ135    orthogonal lifted from D13
ρ622020ζ139+ζ134ζ138+ζ135ζ1310+ζ133ζ1312+ζ13ζ1311+ζ132ζ137+ζ136ζ1310+ζ133ζ139+ζ134ζ1312+ζ13ζ138+ζ135ζ1311+ζ132ζ137+ζ136ζ137+ζ136ζ138+ζ135ζ1310+ζ133ζ1312+ζ13ζ1312+ζ13ζ1310+ζ133ζ138+ζ135ζ137+ζ136ζ139+ζ134ζ1311+ζ132ζ1311+ζ132ζ139+ζ134    orthogonal lifted from D13
ρ722020ζ1310+ζ133ζ137+ζ136ζ1312+ζ13ζ139+ζ134ζ138+ζ135ζ1311+ζ132ζ1312+ζ13ζ1310+ζ133ζ139+ζ134ζ137+ζ136ζ138+ζ135ζ1311+ζ132ζ1311+ζ132ζ137+ζ136ζ1312+ζ13ζ139+ζ134ζ139+ζ134ζ1312+ζ13ζ137+ζ136ζ1311+ζ132ζ1310+ζ133ζ138+ζ135ζ138+ζ135ζ1310+ζ133    orthogonal lifted from D13
ρ822020ζ137+ζ136ζ1312+ζ13ζ1311+ζ132ζ138+ζ135ζ1310+ζ133ζ139+ζ134ζ1311+ζ132ζ137+ζ136ζ138+ζ135ζ1312+ζ13ζ1310+ζ133ζ139+ζ134ζ139+ζ134ζ1312+ζ13ζ1311+ζ132ζ138+ζ135ζ138+ζ135ζ1311+ζ132ζ1312+ζ13ζ139+ζ134ζ137+ζ136ζ1310+ζ133ζ1310+ζ133ζ137+ζ136    orthogonal lifted from D13
ρ922020ζ1312+ζ13ζ1311+ζ132ζ139+ζ134ζ1310+ζ133ζ137+ζ136ζ138+ζ135ζ139+ζ134ζ1312+ζ13ζ1310+ζ133ζ1311+ζ132ζ137+ζ136ζ138+ζ135ζ138+ζ135ζ1311+ζ132ζ139+ζ134ζ1310+ζ133ζ1310+ζ133ζ139+ζ134ζ1311+ζ132ζ138+ζ135ζ1312+ζ13ζ137+ζ136ζ137+ζ136ζ1312+ζ13    orthogonal lifted from D13
ρ10220-10ζ138+ζ135ζ1310+ζ133ζ137+ζ136ζ1311+ζ132ζ139+ζ134ζ1312+ζ13ζ137+ζ136ζ138+ζ135ζ1311+ζ132ζ1310+ζ133ζ139+ζ134ζ1312+ζ13ζ3ζ1312-ζ3ζ13-ζ13ζ32ζ1310-ζ32ζ133-ζ133ζ3ζ137-ζ3ζ136-ζ136ζ3ζ1311-ζ3ζ132-ζ132ζ32ζ1311-ζ32ζ132-ζ132-ζ3ζ137+ζ3ζ136-ζ137-ζ32ζ1310+ζ32ζ133-ζ1310-ζ3ζ1312+ζ3ζ13-ζ1312-ζ3ζ138+ζ3ζ135-ζ138-ζ32ζ139+ζ32ζ134-ζ139ζ32ζ139-ζ32ζ134-ζ134-ζ32ζ138+ζ32ζ135-ζ138    orthogonal lifted from D39
ρ11220-10ζ139+ζ134ζ138+ζ135ζ1310+ζ133ζ1312+ζ13ζ1311+ζ132ζ137+ζ136ζ1310+ζ133ζ139+ζ134ζ1312+ζ13ζ138+ζ135ζ1311+ζ132ζ137+ζ136-ζ3ζ137+ζ3ζ136-ζ137-ζ32ζ138+ζ32ζ135-ζ138-ζ32ζ1310+ζ32ζ133-ζ1310ζ3ζ1312-ζ3ζ13-ζ13-ζ3ζ1312+ζ3ζ13-ζ1312ζ32ζ1310-ζ32ζ133-ζ133-ζ3ζ138+ζ3ζ135-ζ138ζ3ζ137-ζ3ζ136-ζ136-ζ32ζ139+ζ32ζ134-ζ139ζ3ζ1311-ζ3ζ132-ζ132ζ32ζ1311-ζ32ζ132-ζ132ζ32ζ139-ζ32ζ134-ζ134    orthogonal lifted from D39
ρ12220-10ζ1312+ζ13ζ1311+ζ132ζ139+ζ134ζ1310+ζ133ζ137+ζ136ζ138+ζ135ζ139+ζ134ζ1312+ζ13ζ1310+ζ133ζ1311+ζ132ζ137+ζ136ζ138+ζ135-ζ32ζ138+ζ32ζ135-ζ138ζ32ζ1311-ζ32ζ132-ζ132-ζ32ζ139+ζ32ζ134-ζ139ζ32ζ1310-ζ32ζ133-ζ133-ζ32ζ1310+ζ32ζ133-ζ1310ζ32ζ139-ζ32ζ134-ζ134ζ3ζ1311-ζ3ζ132-ζ132-ζ3ζ138+ζ3ζ135-ζ138ζ3ζ1312-ζ3ζ13-ζ13-ζ3ζ137+ζ3ζ136-ζ137ζ3ζ137-ζ3ζ136-ζ136-ζ3ζ1312+ζ3ζ13-ζ1312    orthogonal lifted from D39
ρ13220-10ζ1312+ζ13ζ1311+ζ132ζ139+ζ134ζ1310+ζ133ζ137+ζ136ζ138+ζ135ζ139+ζ134ζ1312+ζ13ζ1310+ζ133ζ1311+ζ132ζ137+ζ136ζ138+ζ135-ζ3ζ138+ζ3ζ135-ζ138ζ3ζ1311-ζ3ζ132-ζ132ζ32ζ139-ζ32ζ134-ζ134-ζ32ζ1310+ζ32ζ133-ζ1310ζ32ζ1310-ζ32ζ133-ζ133-ζ32ζ139+ζ32ζ134-ζ139ζ32ζ1311-ζ32ζ132-ζ132-ζ32ζ138+ζ32ζ135-ζ138-ζ3ζ1312+ζ3ζ13-ζ1312ζ3ζ137-ζ3ζ136-ζ136-ζ3ζ137+ζ3ζ136-ζ137ζ3ζ1312-ζ3ζ13-ζ13    orthogonal lifted from D39
ρ14220-10ζ137+ζ136ζ1312+ζ13ζ1311+ζ132ζ138+ζ135ζ1310+ζ133ζ139+ζ134ζ1311+ζ132ζ137+ζ136ζ138+ζ135ζ1312+ζ13ζ1310+ζ133ζ139+ζ134ζ32ζ139-ζ32ζ134-ζ134-ζ3ζ1312+ζ3ζ13-ζ1312ζ3ζ1311-ζ3ζ132-ζ132-ζ32ζ138+ζ32ζ135-ζ138-ζ3ζ138+ζ3ζ135-ζ138ζ32ζ1311-ζ32ζ132-ζ132ζ3ζ1312-ζ3ζ13-ζ13-ζ32ζ139+ζ32ζ134-ζ139-ζ3ζ137+ζ3ζ136-ζ137ζ32ζ1310-ζ32ζ133-ζ133-ζ32ζ1310+ζ32ζ133-ζ1310ζ3ζ137-ζ3ζ136-ζ136    orthogonal lifted from D39
ρ15220-10ζ1311+ζ132ζ139+ζ134ζ138+ζ135ζ137+ζ136ζ1312+ζ13ζ1310+ζ133ζ138+ζ135ζ1311+ζ132ζ137+ζ136ζ139+ζ134ζ1312+ζ13ζ1310+ζ133ζ32ζ1310-ζ32ζ133-ζ133ζ32ζ139-ζ32ζ134-ζ134-ζ3ζ138+ζ3ζ135-ζ138-ζ3ζ137+ζ3ζ136-ζ137ζ3ζ137-ζ3ζ136-ζ136-ζ32ζ138+ζ32ζ135-ζ138-ζ32ζ139+ζ32ζ134-ζ139-ζ32ζ1310+ζ32ζ133-ζ1310ζ3ζ1311-ζ3ζ132-ζ132ζ3ζ1312-ζ3ζ13-ζ13-ζ3ζ1312+ζ3ζ13-ζ1312ζ32ζ1311-ζ32ζ132-ζ132    orthogonal lifted from D39
ρ16220-10ζ139+ζ134ζ138+ζ135ζ1310+ζ133ζ1312+ζ13ζ1311+ζ132ζ137+ζ136ζ1310+ζ133ζ139+ζ134ζ1312+ζ13ζ138+ζ135ζ1311+ζ132ζ137+ζ136ζ3ζ137-ζ3ζ136-ζ136-ζ3ζ138+ζ3ζ135-ζ138ζ32ζ1310-ζ32ζ133-ζ133-ζ3ζ1312+ζ3ζ13-ζ1312ζ3ζ1312-ζ3ζ13-ζ13-ζ32ζ1310+ζ32ζ133-ζ1310-ζ32ζ138+ζ32ζ135-ζ138-ζ3ζ137+ζ3ζ136-ζ137ζ32ζ139-ζ32ζ134-ζ134ζ32ζ1311-ζ32ζ132-ζ132ζ3ζ1311-ζ3ζ132-ζ132-ζ32ζ139+ζ32ζ134-ζ139    orthogonal lifted from D39
ρ17220-10ζ1311+ζ132ζ139+ζ134ζ138+ζ135ζ137+ζ136ζ1312+ζ13ζ1310+ζ133ζ138+ζ135ζ1311+ζ132ζ137+ζ136ζ139+ζ134ζ1312+ζ13ζ1310+ζ133-ζ32ζ1310+ζ32ζ133-ζ1310-ζ32ζ139+ζ32ζ134-ζ139-ζ32ζ138+ζ32ζ135-ζ138ζ3ζ137-ζ3ζ136-ζ136-ζ3ζ137+ζ3ζ136-ζ137-ζ3ζ138+ζ3ζ135-ζ138ζ32ζ139-ζ32ζ134-ζ134ζ32ζ1310-ζ32ζ133-ζ133ζ32ζ1311-ζ32ζ132-ζ132-ζ3ζ1312+ζ3ζ13-ζ1312ζ3ζ1312-ζ3ζ13-ζ13ζ3ζ1311-ζ3ζ132-ζ132    orthogonal lifted from D39
ρ18220-10ζ1310+ζ133ζ137+ζ136ζ1312+ζ13ζ139+ζ134ζ138+ζ135ζ1311+ζ132ζ1312+ζ13ζ1310+ζ133ζ139+ζ134ζ137+ζ136ζ138+ζ135ζ1311+ζ132ζ3ζ1311-ζ3ζ132-ζ132-ζ3ζ137+ζ3ζ136-ζ137-ζ3ζ1312+ζ3ζ13-ζ1312-ζ32ζ139+ζ32ζ134-ζ139ζ32ζ139-ζ32ζ134-ζ134ζ3ζ1312-ζ3ζ13-ζ13ζ3ζ137-ζ3ζ136-ζ136ζ32ζ1311-ζ32ζ132-ζ132-ζ32ζ1310+ζ32ζ133-ζ1310-ζ3ζ138+ζ3ζ135-ζ138-ζ32ζ138+ζ32ζ135-ζ138ζ32ζ1310-ζ32ζ133-ζ133    orthogonal lifted from D39
ρ19220-10ζ137+ζ136ζ1312+ζ13ζ1311+ζ132ζ138+ζ135ζ1310+ζ133ζ139+ζ134ζ1311+ζ132ζ137+ζ136ζ138+ζ135ζ1312+ζ13ζ1310+ζ133ζ139+ζ134-ζ32ζ139+ζ32ζ134-ζ139ζ3ζ1312-ζ3ζ13-ζ13ζ32ζ1311-ζ32ζ132-ζ132-ζ3ζ138+ζ3ζ135-ζ138-ζ32ζ138+ζ32ζ135-ζ138ζ3ζ1311-ζ3ζ132-ζ132-ζ3ζ1312+ζ3ζ13-ζ1312ζ32ζ139-ζ32ζ134-ζ134ζ3ζ137-ζ3ζ136-ζ136-ζ32ζ1310+ζ32ζ133-ζ1310ζ32ζ1310-ζ32ζ133-ζ133-ζ3ζ137+ζ3ζ136-ζ137    orthogonal lifted from D39
ρ20220-10ζ138+ζ135ζ1310+ζ133ζ137+ζ136ζ1311+ζ132ζ139+ζ134ζ1312+ζ13ζ137+ζ136ζ138+ζ135ζ1311+ζ132ζ1310+ζ133ζ139+ζ134ζ1312+ζ13-ζ3ζ1312+ζ3ζ13-ζ1312-ζ32ζ1310+ζ32ζ133-ζ1310-ζ3ζ137+ζ3ζ136-ζ137ζ32ζ1311-ζ32ζ132-ζ132ζ3ζ1311-ζ3ζ132-ζ132ζ3ζ137-ζ3ζ136-ζ136ζ32ζ1310-ζ32ζ133-ζ133ζ3ζ1312-ζ3ζ13-ζ13-ζ32ζ138+ζ32ζ135-ζ138ζ32ζ139-ζ32ζ134-ζ134-ζ32ζ139+ζ32ζ134-ζ139-ζ3ζ138+ζ3ζ135-ζ138    orthogonal lifted from D39
ρ21220-10ζ1310+ζ133ζ137+ζ136ζ1312+ζ13ζ139+ζ134ζ138+ζ135ζ1311+ζ132ζ1312+ζ13ζ1310+ζ133ζ139+ζ134ζ137+ζ136ζ138+ζ135ζ1311+ζ132ζ32ζ1311-ζ32ζ132-ζ132ζ3ζ137-ζ3ζ136-ζ136ζ3ζ1312-ζ3ζ13-ζ13ζ32ζ139-ζ32ζ134-ζ134-ζ32ζ139+ζ32ζ134-ζ139-ζ3ζ1312+ζ3ζ13-ζ1312-ζ3ζ137+ζ3ζ136-ζ137ζ3ζ1311-ζ3ζ132-ζ132ζ32ζ1310-ζ32ζ133-ζ133-ζ32ζ138+ζ32ζ135-ζ138-ζ3ζ138+ζ3ζ135-ζ138-ζ32ζ1310+ζ32ζ133-ζ1310    orthogonal lifted from D39
ρ223-110-1333333-1-1-1-1-1-1000000000000    orthogonal lifted from S4
ρ233-1-101333333-1-1-1-1-1-1000000000000    orthogonal lifted from S4
ρ246-20003ζ138+3ζ1353ζ1310+3ζ1333ζ137+3ζ1363ζ1311+3ζ1323ζ139+3ζ1343ζ1312+3ζ13-ζ137-ζ136-ζ138-ζ135-ζ1311-ζ132-ζ1310-ζ133-ζ139-ζ134-ζ1312-ζ13000000000000    orthogonal faithful
ρ256-20003ζ1310+3ζ1333ζ137+3ζ1363ζ1312+3ζ133ζ139+3ζ1343ζ138+3ζ1353ζ1311+3ζ132-ζ1312-ζ13-ζ1310-ζ133-ζ139-ζ134-ζ137-ζ136-ζ138-ζ135-ζ1311-ζ132000000000000    orthogonal faithful
ρ266-20003ζ139+3ζ1343ζ138+3ζ1353ζ1310+3ζ1333ζ1312+3ζ133ζ1311+3ζ1323ζ137+3ζ136-ζ1310-ζ133-ζ139-ζ134-ζ1312-ζ13-ζ138-ζ135-ζ1311-ζ132-ζ137-ζ136000000000000    orthogonal faithful
ρ276-20003ζ137+3ζ1363ζ1312+3ζ133ζ1311+3ζ1323ζ138+3ζ1353ζ1310+3ζ1333ζ139+3ζ134-ζ1311-ζ132-ζ137-ζ136-ζ138-ζ135-ζ1312-ζ13-ζ1310-ζ133-ζ139-ζ134000000000000    orthogonal faithful
ρ286-20003ζ1312+3ζ133ζ1311+3ζ1323ζ139+3ζ1343ζ1310+3ζ1333ζ137+3ζ1363ζ138+3ζ135-ζ139-ζ134-ζ1312-ζ13-ζ1310-ζ133-ζ1311-ζ132-ζ137-ζ136-ζ138-ζ135000000000000    orthogonal faithful
ρ296-20003ζ1311+3ζ1323ζ139+3ζ1343ζ138+3ζ1353ζ137+3ζ1363ζ1312+3ζ133ζ1310+3ζ133-ζ138-ζ135-ζ1311-ζ132-ζ137-ζ136-ζ139-ζ134-ζ1312-ζ13-ζ1310-ζ133000000000000    orthogonal faithful

Smallest permutation representation of C13⋊S4
►On 52 points
Generators in S52
(1 2 3 4 5 6 7 8 9 10 11 12 13)(14 15 16 17 18 19 20 21 22 23 24 25 26)(27 28 29 30 31 32 33 34 35 36 37 38 39)(40 41 42 43 44 45 46 47 48 49 50 51 52)

(1 22)(2 23)(3 24)(4 25)(5 26)(6 14)(7 15)(8 16)(9 17)(10 18)(11 19)(12 20)(13 21)(27 44)(28 45)(29 46)(30 47)(31 48)(32 49)(33 50)(34 51)(35 52)(36 40)(37 41)(38 42)(39 43)

(1 28)(2 29)(3 30)(4 31)(5 32)(6 33)(7 34)(8 35)(9 36)(10 37)(11 38)(12 39)(13 27)(14 50)(15 51)(16 52)(17 40)(18 41)(19 42)(20 43)(21 44)(22 45)(23 46)(24 47)(25 48)(26 49)

(14 33 50)(15 34 51)(16 35 52)(17 36 40)(18 37 41)(19 38 42)(20 39 43)(21 27 44)(22 28 45)(23 29 46)(24 30 47)(25 31 48)(26 32 49)

(2 13)(3 12)(4 11)(5 10)(6 9)(7 8)(14 40)(15 52)(16 51)(17 50)(18 49)(19 48)(20 47)(21 46)(22 45)(23 44)(24 43)(25 42)(26 41)(27 29)(30 39)(31 38)(32 37)(33 36)(34 35)
 
G:=sub<Sym(52)| (1,2,3,4,5,6,7,8,9,10,11,12,13)(14,15,16,17,18,19,20,21,22,23,24,25,26)(27,28,29,30,31,32,33,34,35,36,37,38,39)(40,41,42,43,44,45,46,47,48,49,50,51,52), (1,22)(2,23)(3,24)(4,25)(5,26)(6,14)(7,15)(8,16)(9,17)(10,18)(11,19)(12,20)(13,21)(27,44)(28,45)(29,46)(30,47)(31,48)(32,49)(33,50)(34,51)(35,52)(36,40)(37,41)(38,42)(39,43), (1,28)(2,29)(3,30)(4,31)(5,32)(6,33)(7,34)(8,35)(9,36)(10,37)(11,38)(12,39)(13,27)(14,50)(15,51)(16,52)(17,40)(18,41)(19,42)(20,43)(21,44)(22,45)(23,46)(24,47)(25,48)(26,49), (14,33,50)(15,34,51)(16,35,52)(17,36,40)(18,37,41)(19,38,42)(20,39,43)(21,27,44)(22,28,45)(23,29,46)(24,30,47)(25,31,48)(26,32,49), (2,13)(3,12)(4,11)(5,10)(6,9)(7,8)(14,40)(15,52)(16,51)(17,50)(18,49)(19,48)(20,47)(21,46)(22,45)(23,44)(24,43)(25,42)(26,41)(27,29)(30,39)(31,38)(32,37)(33,36)(34,35)>;
 
G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13)(14,15,16,17,18,19,20,21,22,23,24,25,26)(27,28,29,30,31,32,33,34,35,36,37,38,39)(40,41,42,43,44,45,46,47,48,49,50,51,52), (1,22)(2,23)(3,24)(4,25)(5,26)(6,14)(7,15)(8,16)(9,17)(10,18)(11,19)(12,20)(13,21)(27,44)(28,45)(29,46)(30,47)(31,48)(32,49)(33,50)(34,51)(35,52)(36,40)(37,41)(38,42)(39,43), (1,28)(2,29)(3,30)(4,31)(5,32)(6,33)(7,34)(8,35)(9,36)(10,37)(11,38)(12,39)(13,27)(14,50)(15,51)(16,52)(17,40)(18,41)(19,42)(20,43)(21,44)(22,45)(23,46)(24,47)(25,48)(26,49), (14,33,50)(15,34,51)(16,35,52)(17,36,40)(18,37,41)(19,38,42)(20,39,43)(21,27,44)(22,28,45)(23,29,46)(24,30,47)(25,31,48)(26,32,49), (2,13)(3,12)(4,11)(5,10)(6,9)(7,8)(14,40)(15,52)(16,51)(17,50)(18,49)(19,48)(20,47)(21,46)(22,45)(23,44)(24,43)(25,42)(26,41)(27,29)(30,39)(31,38)(32,37)(33,36)(34,35) );
 
G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10,11,12,13),(14,15,16,17,18,19,20,21,22,23,24,25,26),(27,28,29,30,31,32,33,34,35,36,37,38,39),(40,41,42,43,44,45,46,47,48,49,50,51,52)], [(1,22),(2,23),(3,24),(4,25),(5,26),(6,14),(7,15),(8,16),(9,17),(10,18),(11,19),(12,20),(13,21),(27,44),(28,45),(29,46),(30,47),(31,48),(32,49),(33,50),(34,51),(35,52),(36,40),(37,41),(38,42),(39,43)], [(1,28),(2,29),(3,30),(4,31),(5,32),(6,33),(7,34),(8,35),(9,36),(10,37),(11,38),(12,39),(13,27),(14,50),(15,51),(16,52),(17,40),(18,41),(19,42),(20,43),(21,44),(22,45),(23,46),(24,47),(25,48),(26,49)], [(14,33,50),(15,34,51),(16,35,52),(17,36,40),(18,37,41),(19,38,42),(20,39,43),(21,27,44),(22,28,45),(23,29,46),(24,30,47),(25,31,48),(26,32,49)], [(2,13),(3,12),(4,11),(5,10),(6,9),(7,8),(14,40),(15,52),(16,51),(17,50),(18,49),(19,48),(20,47),(21,46),(22,45),(23,44),(24,43),(25,42),(26,41),(27,29),(30,39),(31,38),(32,37),(33,36),(34,35)]])
 

Matrix representation of C13⋊S4 ►in GL5(𝔽157)

458000
9962000
00100
00010
00001
,
10000
01000
0001561
0001560
0011560
,
10000
01000
0001156
0010156
0000156
,
1561000
1560000
00001
00100
00010
,
9558000
15362000
00010
00100
00001

G:=sub<GL(5,GF(157))| [4,99,0,0,0,58,62,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,156,156,156,0,0,1,0,0],[1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,156,156,156],[156,156,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0],[95,153,0,0,0,58,62,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,1] >;
 

C13⋊S4 in GAP, Magma, Sage, TeX 

C_{13}\rtimes S_4
 
% in TeX
 
G:=Group("C13:S4");
 
// GroupNames label
 
G:=SmallGroup(312,48);
 
// by ID
 
G=gap.SmallGroup(312,48);
 
# by ID
 
G:=PCGroup([5,-2,-3,-13,-2,2,41,1082,3123,1568,1954,2934]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^13=b^2=c^2=d^3=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,e*a*e=a^-1,d*b*d^-1=e*b*e=b*c=c*b,d*c*d^-1=b,c*e=e*c,e*d*e=d^-1>;
 
// generators/relations
 

Export

Subgroup lattice of C13⋊S4 in TeX
Character table of C13⋊S4 in TeX

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