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## G = C2×S3×C3⋊D4order 288 = 25·32

### Direct product of C2, S3 and C3⋊D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6 — C2×S3×C3⋊D4
 Chief series C1 — C3 — C32 — C3×C6 — S3×C6 — C2×S32 — C22×S32 — C2×S3×C3⋊D4
 Lower central C32 — C3×C6 — C2×S3×C3⋊D4
 Upper central C1 — C22 — C23

Generators and relations for C2×S3×C3⋊D4
G = < a,b,c,d,e,f | a2=b3=c2=d3=e4=f2=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, cbc=b-1, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, ede-1=fdf=d-1, fef=e-1 >

Subgroups: 1922 in 499 conjugacy classes, 132 normal (36 characteristic)
C1, C2, C2 [×2], C2 [×12], C3 [×2], C3, C4 [×4], C22, C22 [×2], C22 [×36], S3 [×4], S3 [×10], C6 [×2], C6 [×4], C6 [×17], C2×C4 [×6], D4 [×16], C23, C23 [×20], C32, Dic3 [×2], Dic3 [×6], C12 [×2], D6 [×10], D6 [×40], C2×C6 [×2], C2×C6 [×4], C2×C6 [×31], C22×C4, C2×D4 [×12], C24 [×2], C3×S3 [×4], C3×S3 [×4], C3⋊S3 [×2], C3×C6, C3×C6 [×2], C3×C6 [×2], C4×S3 [×4], D12 [×4], C2×Dic3, C2×Dic3 [×7], C3⋊D4 [×4], C3⋊D4 [×24], C2×C12, C3×D4 [×4], C22×S3 [×3], C22×S3 [×4], C22×S3 [×23], C22×C6 [×2], C22×C6 [×12], C22×D4, C3×Dic3 [×2], C3⋊Dic3 [×2], S32 [×8], S3×C6 [×10], S3×C6 [×12], C2×C3⋊S3 [×2], C2×C3⋊S3 [×2], C62, C62 [×2], C62 [×2], S3×C2×C4, C2×D12, S3×D4 [×8], C22×Dic3, C2×C3⋊D4, C2×C3⋊D4 [×14], C6×D4, S3×C23, S3×C23 [×2], C23×C6, S3×Dic3 [×4], D6⋊S3 [×4], C3⋊D12 [×4], C6×Dic3, C3×C3⋊D4 [×4], C2×C3⋊Dic3, C327D4 [×4], C2×S32 [×4], C2×S32 [×4], S3×C2×C6 [×3], S3×C2×C6 [×4], S3×C2×C6 [×4], C22×C3⋊S3, C2×C62, C2×S3×D4, C22×C3⋊D4, C2×S3×Dic3, C2×D6⋊S3, C2×C3⋊D12, S3×C3⋊D4 [×8], C6×C3⋊D4, C2×C327D4, C22×S32, S3×C22×C6, C2×S3×C3⋊D4
Quotients: C1, C2 [×15], C22 [×35], S3 [×2], D4 [×4], C23 [×15], D6 [×14], C2×D4 [×6], C24, C3⋊D4 [×4], C22×S3 [×14], C22×D4, S32, S3×D4 [×2], C2×C3⋊D4 [×6], S3×C23 [×2], C2×S32 [×3], C2×S3×D4, C22×C3⋊D4, S3×C3⋊D4 [×2], C22×S32, C2×S3×C3⋊D4

Smallest permutation representation of C2×S3×C3⋊D4
On 48 points
Generators in S48
(1 30)(2 31)(3 32)(4 29)(5 35)(6 36)(7 33)(8 34)(9 42)(10 43)(11 44)(12 41)(13 24)(14 21)(15 22)(16 23)(17 27)(18 28)(19 25)(20 26)(37 47)(38 48)(39 45)(40 46)
(1 44 45)(2 41 46)(3 42 47)(4 43 48)(5 22 26)(6 23 27)(7 24 28)(8 21 25)(9 37 32)(10 38 29)(11 39 30)(12 40 31)(13 18 33)(14 19 34)(15 20 35)(16 17 36)
(1 6)(2 7)(3 8)(4 5)(9 19)(10 20)(11 17)(12 18)(13 40)(14 37)(15 38)(16 39)(21 47)(22 48)(23 45)(24 46)(25 42)(26 43)(27 44)(28 41)(29 35)(30 36)(31 33)(32 34)
(1 45 44)(2 41 46)(3 47 42)(4 43 48)(5 26 22)(6 23 27)(7 28 24)(8 21 25)(9 32 37)(10 38 29)(11 30 39)(12 40 31)(13 33 18)(14 19 34)(15 35 20)(16 17 36)
(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)
(1 34)(2 33)(3 36)(4 35)(5 29)(6 32)(7 31)(8 30)(9 23)(10 22)(11 21)(12 24)(13 41)(14 44)(15 43)(16 42)(17 47)(18 46)(19 45)(20 48)(25 39)(26 38)(27 37)(28 40)

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

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

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

54 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 2J 2K 2L 2M 2N 2O 3A 3B 3C 4A 4B 4C 4D 6A ··· 6J 6K ··· 6S 6T ··· 6AA 6AB 6AC 12A 12B order 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3 3 3 4 4 4 4 6 ··· 6 6 ··· 6 6 ··· 6 6 6 12 12 size 1 1 1 1 2 2 3 3 3 3 6 6 6 6 18 18 2 2 4 6 6 18 18 2 ··· 2 4 ··· 4 6 ··· 6 12 12 12 12

54 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C2 S3 S3 D4 D6 D6 D6 D6 C3⋊D4 S32 S3×D4 C2×S32 S3×C3⋊D4 kernel C2×S3×C3⋊D4 C2×S3×Dic3 C2×D6⋊S3 C2×C3⋊D12 S3×C3⋊D4 C6×C3⋊D4 C2×C32⋊7D4 C22×S32 S3×C22×C6 C2×C3⋊D4 S3×C23 S3×C6 C2×Dic3 C3⋊D4 C22×S3 C22×C6 D6 C23 C6 C22 C2 # reps 1 1 1 1 8 1 1 1 1 1 1 4 1 4 7 2 8 1 2 3 4

Matrix representation of C2×S3×C3⋊D4 in GL8(ℤ)

 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 -1 1 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 -1
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 -1 1 0 0 0 0 0 0 -1 0
,
 -1 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 -1 0
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 -1 0

G:=sub<GL(8,Integers())| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[-1,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,-1,0,0,0,0,0,0,1,0],[-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0] >;

C2×S3×C3⋊D4 in GAP, Magma, Sage, TeX

C_2\times S_3\times C_3\rtimes D_4
% in TeX

G:=Group("C2xS3xC3:D4");
// GroupNames label

G:=SmallGroup(288,976);
// by ID

G=gap.SmallGroup(288,976);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,346,1356,9414]);
// Polycyclic

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

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