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## G = (C22×S3)⋊C8order 192 = 26·3

### The semidirect product of C22×S3 and C8 acting via C8/C2=C4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — (C22×S3)⋊C8
 Chief series C1 — C3 — C6 — C2×C6 — C2×C12 — C22×C12 — C2×D6⋊C4 — (C22×S3)⋊C8
 Lower central C3 — C6 — C2×C6 — (C22×S3)⋊C8
 Upper central C1 — C22 — C22×C4 — C22⋊C8

Generators and relations for (C22×S3)⋊C8
G = < a,b,c,d,e | a2=b2=c3=d2=e8=1, eae-1=ab=ba, ac=ca, ede-1=ad=da, bc=cb, bd=db, be=eb, dcd=c-1, ce=ec >

Subgroups: 368 in 98 conjugacy classes, 31 normal (29 characteristic)
C1, C2, C2, C3, C4, C22, C22, S3, C6, C6, C8, C2×C4, C2×C4, C23, C23, Dic3, C12, D6, C2×C6, C2×C6, C22⋊C4, C2×C8, C22×C4, C22×C4, C24, C3⋊C8, C24, C2×Dic3, C2×C12, C2×C12, C22×S3, C22×S3, C22×C6, C22⋊C8, C22⋊C8, C2×C22⋊C4, C2×C3⋊C8, D6⋊C4, C2×C24, C22×Dic3, C22×C12, S3×C23, C23⋊C8, C12.55D4, C3×C22⋊C8, C2×D6⋊C4, (C22×S3)⋊C8
Quotients: C1, C2, C4, C22, S3, C8, C2×C4, D4, D6, C22⋊C4, C2×C8, M4(2), C4×S3, D12, C3⋊D4, C22⋊C8, C23⋊C4, C4.D4, S3×C8, C8⋊S3, D6⋊C4, C23⋊C8, C23.6D6, D6⋊C8, C12.46D4, (C22×S3)⋊C8

Smallest permutation representation of (C22×S3)⋊C8
On 48 points
Generators in S48
(1 5)(2 25)(3 7)(4 27)(6 29)(8 31)(9 47)(10 14)(11 41)(12 16)(13 43)(15 45)(17 33)(18 22)(19 35)(20 24)(21 37)(23 39)(26 30)(28 32)(34 38)(36 40)(42 46)(44 48)
(1 28)(2 29)(3 30)(4 31)(5 32)(6 25)(7 26)(8 27)(9 43)(10 44)(11 45)(12 46)(13 47)(14 48)(15 41)(16 42)(17 37)(18 38)(19 39)(20 40)(21 33)(22 34)(23 35)(24 36)
(1 16 24)(2 9 17)(3 10 18)(4 11 19)(5 12 20)(6 13 21)(7 14 22)(8 15 23)(25 47 33)(26 48 34)(27 41 35)(28 42 36)(29 43 37)(30 44 38)(31 45 39)(32 46 40)
(1 28)(2 25)(4 8)(5 32)(6 29)(9 33)(10 18)(11 23)(12 40)(13 37)(14 22)(15 19)(16 36)(17 47)(20 46)(21 43)(24 42)(27 31)(34 48)(35 45)(38 44)(39 41)
(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)

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

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

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

42 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3 4A 4B 4C 4D 4E 4F 6A 6B 6C 6D 6E 8A 8B 8C 8D 8E 8F 8G 8H 12A 12B 12C 12D 12E 12F 24A ··· 24H order 1 2 2 2 2 2 2 2 3 4 4 4 4 4 4 6 6 6 6 6 8 8 8 8 8 8 8 8 12 12 12 12 12 12 24 ··· 24 size 1 1 1 1 2 2 12 12 2 2 2 2 2 12 12 2 2 2 4 4 4 4 4 4 12 12 12 12 2 2 2 2 4 4 4 ··· 4

42 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + image C1 C2 C2 C2 C4 C4 C8 S3 D4 D6 M4(2) D12 C3⋊D4 C4×S3 S3×C8 C8⋊S3 C23⋊C4 C4.D4 C23.6D6 C12.46D4 kernel (C22×S3)⋊C8 C12.55D4 C3×C22⋊C8 C2×D6⋊C4 C22×Dic3 S3×C23 C22×S3 C22⋊C8 C2×C12 C22×C4 C2×C6 C2×C4 C2×C4 C23 C22 C22 C6 C6 C2 C2 # reps 1 1 1 1 2 2 8 1 2 1 2 2 2 2 4 4 1 1 2 2

Matrix representation of (C22×S3)⋊C8 in GL6(𝔽73)

 72 0 0 0 0 0 0 72 0 0 0 0 0 0 72 0 0 0 0 0 0 72 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 72 0 0 0 0 0 0 72 0 0 0 0 0 0 72 0 0 0 0 0 0 72
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 72 1 0 0 0 0 72 0 0 0 0 0 0 0 72 1 0 0 0 0 72 0
,
 1 0 0 0 0 0 29 72 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0
,
 22 1 0 0 0 0 54 51 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 66 14 0 0 0 0 59 7 0 0

G:=sub<GL(6,GF(73))| [72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,72,0,0,0,0,1,0,0,0,0,0,0,0,72,72,0,0,0,0,1,0],[1,29,0,0,0,0,0,72,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[22,54,0,0,0,0,1,51,0,0,0,0,0,0,0,0,66,59,0,0,0,0,14,7,0,0,1,0,0,0,0,0,0,1,0,0] >;

(C22×S3)⋊C8 in GAP, Magma, Sage, TeX

(C_2^2\times S_3)\rtimes C_8
% in TeX

G:=Group("(C2^2xS3):C8");
// GroupNames label

G:=SmallGroup(192,27);
// by ID

G=gap.SmallGroup(192,27);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,141,36,758,100,570,6278]);
// Polycyclic

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

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