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G = C8.7S4order 192 = 26·3

2nd central extension by C8 of S4

non-abelian, soluble

Aliases: C8.7S4, SL2(𝔽3).C8, Q8.(C3⋊C8), C8○D4.2S3, C4.A4.1C4, C8.A4.2C2, C2.3(A4⋊C8), C4.5(A4⋊C4), C4○D4.1Dic3, SmallGroup(192,187)

Series: Derived Chief Lower central Upper central

C1C2Q8SL2(𝔽3) — C8.7S4
C1C2Q8SL2(𝔽3)C4.A4C8.A4 — C8.7S4
SL2(𝔽3) — C8.7S4
C1C8

Generators and relations for C8.7S4
 G = < a,b,c,d,e | a8=d3=1, b2=c2=a4, e2=a, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=a4b, dbd-1=a4bc, ebe-1=bc, dcd-1=b, ece-1=a4c, ede-1=d-1 >

6C2
4C3
3C22
3C4
4C6
3D4
3C8
3C2×C4
4C12
3C2×C8
3M4(2)
6C16
6C16
4C24
3C2×C16
3M5(2)
4C3⋊C16
3D4.C8

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

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

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

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

32 conjugacy classes

class 1 2A2B 3 4A4B4C 6 8A8B8C8D8E8F12A12B16A···16H16I16J16K16L24A24B24C24D
order12234446888888121216···161616161624242424
size11681168111166886···6121212128888

32 irreducible representations

dim111122223334
type+++-+
imageC1C2C4C8S3Dic3C3⋊C8C8.7S4S4A4⋊C4A4⋊C8C8.7S4
kernelC8.7S4C8.A4C4.A4SL2(𝔽3)C8○D4C4○D4Q8C1C8C4C2C1
# reps112411282244

Matrix representation of C8.7S4 in GL2(𝔽17) generated by

80
08
,
161
151
,
413
013
,
010
516
,
78
1410
G:=sub<GL(2,GF(17))| [8,0,0,8],[16,15,1,1],[4,0,13,13],[0,5,10,16],[7,14,8,10] >;

C8.7S4 in GAP, Magma, Sage, TeX

C_8._7S_4
% in TeX

G:=Group("C8.7S4");
// GroupNames label

G:=SmallGroup(192,187);
// by ID

G=gap.SmallGroup(192,187);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,2,-2,14,36,520,451,1684,655,172,1013,404,285,124]);
// Polycyclic

G:=Group<a,b,c,d,e|a^8=d^3=1,b^2=c^2=a^4,e^2=a,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,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

Subgroup lattice of C8.7S4 in TeX

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