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## G = C2×C3⋊S3⋊3C8order 288 = 25·32

### Direct product of C2 and C3⋊S3⋊3C8

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C2×C3⋊S3⋊3C8
 Chief series C1 — C32 — C3×C6 — C3⋊Dic3 — C32⋊2C8 — C2×C32⋊2C8 — C2×C3⋊S3⋊3C8
 Lower central C32 — C2×C3⋊S3⋊3C8
 Upper central C1 — C2×C4

Generators and relations for C2×C3⋊S33C8
G = < a,b,c,d,e | a2=b3=c3=d2=e8=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, dbd=b-1, ebe-1=bc-1, dcd=c-1, ece-1=b-1c-1, de=ed >

Subgroups: 576 in 130 conjugacy classes, 46 normal (14 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, S3, C6, C8, C2×C4, C2×C4, C23, C32, Dic3, C12, D6, C2×C6, C2×C8, C22×C4, C3⋊S3, C3×C6, C3×C6, C4×S3, C2×Dic3, C2×C12, C22×S3, C22×C8, C3⋊Dic3, C3×C12, C2×C3⋊S3, C62, S3×C2×C4, C322C8, C4×C3⋊S3, C2×C3⋊Dic3, C6×C12, C22×C3⋊S3, C3⋊S33C8, C2×C322C8, C2×C4×C3⋊S3, C2×C3⋊S33C8
Quotients: C1, C2, C4, C22, C8, C2×C4, C23, C2×C8, C22×C4, C22×C8, C32⋊C4, C2×C32⋊C4, C3⋊S33C8, C22×C32⋊C4, C2×C3⋊S33C8

Smallest permutation representation of C2×C3⋊S33C8
On 48 points
Generators in S48
(1 13)(2 14)(3 15)(4 16)(5 9)(6 10)(7 11)(8 12)(17 41)(18 42)(19 43)(20 44)(21 45)(22 46)(23 47)(24 48)(25 33)(26 34)(27 35)(28 36)(29 37)(30 38)(31 39)(32 40)
(1 31 21)(2 32 22)(3 23 25)(4 24 26)(5 27 17)(6 28 18)(7 19 29)(8 20 30)(9 35 41)(10 36 42)(11 43 37)(12 44 38)(13 39 45)(14 40 46)(15 47 33)(16 48 34)
(2 22 32)(4 26 24)(6 18 28)(8 30 20)(10 42 36)(12 38 44)(14 46 40)(16 34 48)
(17 27)(18 28)(19 29)(20 30)(21 31)(22 32)(23 25)(24 26)(33 47)(34 48)(35 41)(36 42)(37 43)(38 44)(39 45)(40 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)

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

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

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

48 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 4A 4B 4C 4D 4E 4F 4G 4H 6A ··· 6F 8A ··· 8P 12A ··· 12H order 1 2 2 2 2 2 2 2 3 3 4 4 4 4 4 4 4 4 6 ··· 6 8 ··· 8 12 ··· 12 size 1 1 1 1 9 9 9 9 4 4 1 1 1 1 9 9 9 9 4 ··· 4 9 ··· 9 4 ··· 4

48 irreducible representations

 dim 1 1 1 1 1 1 1 1 4 4 4 4 type + + + + + + + image C1 C2 C2 C2 C4 C4 C4 C8 C32⋊C4 C2×C32⋊C4 C2×C32⋊C4 C3⋊S3⋊3C8 kernel C2×C3⋊S3⋊3C8 C3⋊S3⋊3C8 C2×C32⋊2C8 C2×C4×C3⋊S3 C4×C3⋊S3 C6×C12 C22×C3⋊S3 C2×C3⋊S3 C2×C4 C4 C22 C2 # reps 1 4 2 1 4 2 2 16 2 4 2 8

Matrix representation of C2×C3⋊S33C8 in GL5(𝔽73)

 72 0 0 0 0 0 72 0 0 0 0 0 72 0 0 0 0 0 72 0 0 0 0 0 72
,
 1 0 0 0 0 0 0 72 0 0 0 1 72 0 0 0 0 0 72 1 0 0 0 72 0
,
 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 72 0 0 0 1 72
,
 72 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 1 0
,
 72 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 27 0 0 0 27 0 0 0

G:=sub<GL(5,GF(73))| [72,0,0,0,0,0,72,0,0,0,0,0,72,0,0,0,0,0,72,0,0,0,0,0,72],[1,0,0,0,0,0,0,1,0,0,0,72,72,0,0,0,0,0,72,72,0,0,0,1,0],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,72,72],[72,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,1,0],[72,0,0,0,0,0,0,0,0,27,0,0,0,27,0,0,1,0,0,0,0,0,1,0,0] >;

C2×C3⋊S33C8 in GAP, Magma, Sage, TeX

C_2\times C_3\rtimes S_3\rtimes_3C_8
% in TeX

G:=Group("C2xC3:S3:3C8");
// GroupNames label

G:=SmallGroup(288,929);
// by ID

G=gap.SmallGroup(288,929);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,56,100,80,9413,362,12550,1203]);
// Polycyclic

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

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