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G = C8.S4order 192 = 26·3

2nd non-split extension by C8 of S4 acting via S4/A4=C2

non-abelian, soluble

Aliases: C8.2S4, Q8.3D12, SL2(F3).3D4, C2.9(C4:S4), C4.19(C2xS4), C4.S4.C2, C8oD4.1S3, C8.A4.1C2, C4oD4.8D6, C4.A4.7C22, SmallGroup(192,962)

Series: Derived Chief Lower central Upper central

C1C2Q8C4.A4 — C8.S4
C1C2Q8SL2(F3)C4.A4C4.S4 — C8.S4
SL2(F3)C4.A4 — C8.S4
C1C2C4C8

Generators and relations for C8.S4
 G = < a,b,c,d,e | a8=d3=1, b2=c2=e2=a4, ab=ba, ac=ca, ad=da, eae-1=a-1, cbc-1=a4b, dbd-1=a4bc, ebe-1=bc, dcd-1=b, ece-1=a4c, ede-1=d-1 >

Subgroups: 229 in 59 conjugacy classes, 13 normal (11 characteristic)
C1, C2, C2, C3, C4, C4, C22, C6, C8, C8, C2xC4, D4, Q8, Q8, Dic3, C12, C2xC8, M4(2), SD16, Q16, C2xQ8, C4oD4, C24, SL2(F3), Dic6, C4.10D4, C8.C4, C8oD4, C2xQ16, C8.C22, Dic12, CSU2(F3), C4.A4, D4.5D4, C8.A4, C4.S4, C8.S4
Quotients: C1, C2, C22, S3, D4, D6, D12, S4, C2xS4, C4:S4, C8.S4

Character table of C8.S4

 class 12A2B34A4B4C4D68A8B8C8D8E12A12B24A24B24C24D
 size 1168262424822122424888888
ρ111111111111111111111    trivial
ρ2111111-1-11111-1-1111111    linear of order 2
ρ31111111-11-1-1-11-111-1-1-1-1    linear of order 2
ρ4111111-111-1-1-1-1111-1-1-1-1    linear of order 2
ρ522-22-2200200000-2-20000    orthogonal lifted from D4
ρ6222-12200-1-2-2-200-1-11111    orthogonal lifted from D6
ρ7222-12200-122200-1-1-1-1-1-1    orthogonal lifted from S3
ρ822-2-1-2200-100000113-33-3    orthogonal lifted from D12
ρ922-2-1-2200-10000011-33-33    orthogonal lifted from D12
ρ1033-103-1-110-3-311-1000000    orthogonal lifted from C2xS4
ρ1133-103-11-10-3-31-11000000    orthogonal lifted from C2xS4
ρ1233-103-111033-1-1-1000000    orthogonal lifted from S4
ρ1333-103-1-1-1033-111000000    orthogonal lifted from S4
ρ144-40-200002-222200000-2-222    symplectic faithful, Schur index 2
ρ154-40-20000222-220000022-2-2    symplectic faithful, Schur index 2
ρ164-4010000-1-2222000-33ζ87ζ38785ζ3ζ83ζ38ζ38ζ87ζ3285ζ3285ζ83ζ32838ζ32    symplectic faithful, Schur index 2
ρ174-4010000-122-220003-3ζ83ζ32838ζ32ζ87ζ3285ζ3285ζ83ζ38ζ38ζ87ζ38785ζ3    symplectic faithful, Schur index 2
ρ184-4010000-122-22000-33ζ87ζ3285ζ3285ζ83ζ32838ζ32ζ87ζ38785ζ3ζ83ζ38ζ38    symplectic faithful, Schur index 2
ρ194-4010000-1-22220003-3ζ83ζ38ζ38ζ87ζ38785ζ3ζ83ζ32838ζ32ζ87ζ3285ζ3285    symplectic faithful, Schur index 2
ρ206620-6-200000000000000    orthogonal lifted from C4:S4

Smallest permutation representation of C8.S4
On 64 points
Generators in S64
(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 53 54 55 56)(57 58 59 60 61 62 63 64)
(1 53 5 49)(2 54 6 50)(3 55 7 51)(4 56 8 52)(9 61 13 57)(10 62 14 58)(11 63 15 59)(12 64 16 60)(17 31 21 27)(18 32 22 28)(19 25 23 29)(20 26 24 30)(33 43 37 47)(34 44 38 48)(35 45 39 41)(36 46 40 42)
(1 35 5 39)(2 36 6 40)(3 37 7 33)(4 38 8 34)(9 32 13 28)(10 25 14 29)(11 26 15 30)(12 27 16 31)(17 64 21 60)(18 57 22 61)(19 58 23 62)(20 59 24 63)(41 49 45 53)(42 50 46 54)(43 51 47 55)(44 52 48 56)
(17 64 27)(18 57 28)(19 58 29)(20 59 30)(21 60 31)(22 61 32)(23 62 25)(24 63 26)(33 43 51)(34 44 52)(35 45 53)(36 46 54)(37 47 55)(38 48 56)(39 41 49)(40 42 50)
(1 15 5 11)(2 14 6 10)(3 13 7 9)(4 12 8 16)(17 56 21 52)(18 55 22 51)(19 54 23 50)(20 53 24 49)(25 40 29 36)(26 39 30 35)(27 38 31 34)(28 37 32 33)(41 59 45 63)(42 58 46 62)(43 57 47 61)(44 64 48 60)

G:=sub<Sym(64)| (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)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64), (1,53,5,49)(2,54,6,50)(3,55,7,51)(4,56,8,52)(9,61,13,57)(10,62,14,58)(11,63,15,59)(12,64,16,60)(17,31,21,27)(18,32,22,28)(19,25,23,29)(20,26,24,30)(33,43,37,47)(34,44,38,48)(35,45,39,41)(36,46,40,42), (1,35,5,39)(2,36,6,40)(3,37,7,33)(4,38,8,34)(9,32,13,28)(10,25,14,29)(11,26,15,30)(12,27,16,31)(17,64,21,60)(18,57,22,61)(19,58,23,62)(20,59,24,63)(41,49,45,53)(42,50,46,54)(43,51,47,55)(44,52,48,56), (17,64,27)(18,57,28)(19,58,29)(20,59,30)(21,60,31)(22,61,32)(23,62,25)(24,63,26)(33,43,51)(34,44,52)(35,45,53)(36,46,54)(37,47,55)(38,48,56)(39,41,49)(40,42,50), (1,15,5,11)(2,14,6,10)(3,13,7,9)(4,12,8,16)(17,56,21,52)(18,55,22,51)(19,54,23,50)(20,53,24,49)(25,40,29,36)(26,39,30,35)(27,38,31,34)(28,37,32,33)(41,59,45,63)(42,58,46,62)(43,57,47,61)(44,64,48,60)>;

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)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64), (1,53,5,49)(2,54,6,50)(3,55,7,51)(4,56,8,52)(9,61,13,57)(10,62,14,58)(11,63,15,59)(12,64,16,60)(17,31,21,27)(18,32,22,28)(19,25,23,29)(20,26,24,30)(33,43,37,47)(34,44,38,48)(35,45,39,41)(36,46,40,42), (1,35,5,39)(2,36,6,40)(3,37,7,33)(4,38,8,34)(9,32,13,28)(10,25,14,29)(11,26,15,30)(12,27,16,31)(17,64,21,60)(18,57,22,61)(19,58,23,62)(20,59,24,63)(41,49,45,53)(42,50,46,54)(43,51,47,55)(44,52,48,56), (17,64,27)(18,57,28)(19,58,29)(20,59,30)(21,60,31)(22,61,32)(23,62,25)(24,63,26)(33,43,51)(34,44,52)(35,45,53)(36,46,54)(37,47,55)(38,48,56)(39,41,49)(40,42,50), (1,15,5,11)(2,14,6,10)(3,13,7,9)(4,12,8,16)(17,56,21,52)(18,55,22,51)(19,54,23,50)(20,53,24,49)(25,40,29,36)(26,39,30,35)(27,38,31,34)(28,37,32,33)(41,59,45,63)(42,58,46,62)(43,57,47,61)(44,64,48,60) );

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),(49,50,51,52,53,54,55,56),(57,58,59,60,61,62,63,64)], [(1,53,5,49),(2,54,6,50),(3,55,7,51),(4,56,8,52),(9,61,13,57),(10,62,14,58),(11,63,15,59),(12,64,16,60),(17,31,21,27),(18,32,22,28),(19,25,23,29),(20,26,24,30),(33,43,37,47),(34,44,38,48),(35,45,39,41),(36,46,40,42)], [(1,35,5,39),(2,36,6,40),(3,37,7,33),(4,38,8,34),(9,32,13,28),(10,25,14,29),(11,26,15,30),(12,27,16,31),(17,64,21,60),(18,57,22,61),(19,58,23,62),(20,59,24,63),(41,49,45,53),(42,50,46,54),(43,51,47,55),(44,52,48,56)], [(17,64,27),(18,57,28),(19,58,29),(20,59,30),(21,60,31),(22,61,32),(23,62,25),(24,63,26),(33,43,51),(34,44,52),(35,45,53),(36,46,54),(37,47,55),(38,48,56),(39,41,49),(40,42,50)], [(1,15,5,11),(2,14,6,10),(3,13,7,9),(4,12,8,16),(17,56,21,52),(18,55,22,51),(19,54,23,50),(20,53,24,49),(25,40,29,36),(26,39,30,35),(27,38,31,34),(28,37,32,33),(41,59,45,63),(42,58,46,62),(43,57,47,61),(44,64,48,60)]])

Matrix representation of C8.S4 in GL4(F7) generated by

1405
6166
5554
4351
,
3206
3245
6420
2310
,
0232
2344
5110
4653
,
5231
0212
3210
2654
,
6620
6014
2352
1413
G:=sub<GL(4,GF(7))| [1,6,5,4,4,1,5,3,0,6,5,5,5,6,4,1],[3,3,6,2,2,2,4,3,0,4,2,1,6,5,0,0],[0,2,5,4,2,3,1,6,3,4,1,5,2,4,0,3],[5,0,3,2,2,2,2,6,3,1,1,5,1,2,0,4],[6,6,2,1,6,0,3,4,2,1,5,1,0,4,2,3] >;

C8.S4 in GAP, Magma, Sage, TeX

C_8.S_4
% in TeX

G:=Group("C8.S4");
// GroupNames label

G:=SmallGroup(192,962);
// by ID

G=gap.SmallGroup(192,962);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,2,-2,672,85,708,2102,520,451,1684,655,172,1013,404,285,124]);
// Polycyclic

G:=Group<a,b,c,d,e|a^8=d^3=1,b^2=c^2=e^2=a^4,a*b=b*a,a*c=c*a,a*d=d*a,e*a*e^-1=a^-1,c*b*c^-1=a^4*b,d*b*d^-1=a^4*b*c,e*b*e^-1=b*c,d*c*d^-1=b,e*c*e^-1=a^4*c,e*d*e^-1=d^-1>;
// generators/relations

Export

Character table of C8.S4 in TeX

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