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## G = C2×C32⋊4S4order 432 = 24·33

### Direct product of C2 and C32⋊4S4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C32×A4 — C2×C32⋊4S4
 Chief series C1 — C22 — C2×C6 — C62 — C32×A4 — C32⋊4S4 — C2×C32⋊4S4
 Lower central C32×A4 — C2×C32⋊4S4
 Upper central C1 — C2

Generators and relations for C2×C324S4
G = < a,b,c,d,e | a2=b6=c6=d3=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, dbd-1=b4c3, ebe=b2c3, dcd-1=b3c, ece=b3c2, ede=d-1 >

Subgroups: 3340 in 358 conjugacy classes, 71 normal (11 characteristic)
C1, C2, C2 [×4], C3 [×4], C3 [×9], C4 [×2], C22, C22 [×6], S3 [×26], C6 [×4], C6 [×17], C2×C4, D4 [×4], C23, C23, C32, C32 [×12], Dic3 [×8], A4 [×9], D6 [×25], C2×C6 [×4], C2×C6 [×8], C2×D4, C3⋊S3 [×26], C3×C6, C3×C6 [×14], C2×Dic3 [×4], C3⋊D4 [×16], S4 [×18], C2×A4 [×9], C22×S3 [×4], C22×C6 [×4], C33, C3⋊Dic3 [×2], C3×A4 [×12], C2×C3⋊S3 [×16], C62, C62 [×2], C2×C3⋊D4 [×4], C2×S4 [×9], C33⋊C2 [×2], C32×C6, C2×C3⋊Dic3, C327D4 [×4], C3⋊S4 [×24], C6×A4 [×12], C22×C3⋊S3, C2×C62, C32×A4, C2×C33⋊C2, C2×C327D4, C2×C3⋊S4 [×12], C324S4 [×2], A4×C3×C6, C2×C324S4
Quotients: C1, C2 [×3], C22, S3 [×13], D6 [×13], C3⋊S3 [×13], S4, C2×C3⋊S3 [×13], C2×S4, C33⋊C2, C3⋊S4 [×4], C2×C33⋊C2, C2×C3⋊S4 [×4], C324S4, C2×C324S4

Smallest permutation representation of C2×C324S4
On 54 points
Generators in S54
(1 8)(2 9)(3 7)(4 13)(5 14)(6 15)(10 17)(11 18)(12 16)(19 22)(20 23)(21 24)(25 28)(26 29)(27 30)(31 34)(32 35)(33 36)(37 40)(38 41)(39 42)(43 46)(44 47)(45 48)(49 52)(50 53)(51 54)
(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)
(1 15 17 8 6 10)(2 13 18 9 4 11)(3 14 16 7 5 12)(19 26 45 22 29 48)(20 27 46 23 30 43)(21 28 47 24 25 44)(31 37 54)(32 38 49)(33 39 50)(34 40 51)(35 41 52)(36 42 53)
(1 46 39)(2 44 37)(3 48 41)(4 24 54)(5 22 52)(6 20 50)(7 45 38)(8 43 42)(9 47 40)(10 27 36)(11 25 34)(12 29 32)(13 21 51)(14 19 49)(15 23 53)(16 26 35)(17 30 33)(18 28 31)
(1 53)(2 49)(3 51)(4 38)(5 40)(6 42)(7 54)(8 50)(9 52)(10 33)(11 35)(12 31)(13 41)(14 37)(15 39)(16 34)(17 36)(18 32)(19 44)(20 43)(21 48)(22 47)(23 46)(24 45)(25 26)(27 30)(28 29)

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

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

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

42 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3A 3B 3C 3D 3E ··· 3M 4A 4B 6A 6B 6C 6D 6E ··· 6L 6M ··· 6U order 1 2 2 2 2 2 3 3 3 3 3 ··· 3 4 4 6 6 6 6 6 ··· 6 6 ··· 6 size 1 1 3 3 54 54 2 2 2 2 8 ··· 8 54 54 2 2 2 2 6 ··· 6 8 ··· 8

42 irreducible representations

 dim 1 1 1 2 2 2 2 3 3 6 6 type + + + + + + + + + + + image C1 C2 C2 S3 S3 D6 D6 S4 C2×S4 C3⋊S4 C2×C3⋊S4 kernel C2×C32⋊4S4 C32⋊4S4 A4×C3×C6 C6×A4 C2×C62 C3×A4 C62 C3×C6 C32 C6 C3 # reps 1 2 1 12 1 12 1 2 2 4 4

Matrix representation of C2×C324S4 in GL7(ℤ)

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

G:=sub<GL(7,Integers())| [-1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,-1],[0,1,0,0,0,0,0,-1,-1,0,0,0,0,0,0,0,-1,-1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,1],[0,1,0,0,0,0,0,-1,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,-1,-1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,-1],[0,1,0,0,0,0,0,-1,-1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0],[1,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,-1,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,-1,0] >;

C2×C324S4 in GAP, Magma, Sage, TeX

C_2\times C_3^2\rtimes_4S_4
% in TeX

G:=Group("C2xC3^2:4S4");
// GroupNames label

G:=SmallGroup(432,762);
// by ID

G=gap.SmallGroup(432,762);
# by ID

G:=PCGroup([7,-2,-2,-3,-3,-3,-2,2,170,675,2524,9077,2287,5298,3989]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^6=c^6=d^3=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,d*b*d^-1=b^4*c^3,e*b*e=b^2*c^3,d*c*d^-1=b^3*c,e*c*e=b^3*c^2,e*d*e=d^-1>;
// generators/relations

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