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G = C2×C324S4order 432 = 24·33

Direct product of C2 and C324S4

direct product, non-abelian, soluble, monomial, rational

Aliases: C2×C324S4, C6219D6, C6⋊(C3⋊S4), (C3×C6)⋊4S4, (C6×A4)⋊4S3, (C3×A4)⋊10D6, (C2×C62)⋊9S3, C329(C2×S4), C23⋊(C33⋊C2), (C32×A4)⋊8C22, C32(C2×C3⋊S4), (A4×C3×C6)⋊3C2, (C2×A4)⋊(C3⋊S3), A42(C2×C3⋊S3), C22⋊(C2×C33⋊C2), (C22×C6)⋊2(C3⋊S3), (C2×C6)⋊3(C2×C3⋊S3), SmallGroup(432,762)

Series: Derived Chief Lower central Upper central

C1C22C32×A4 — C2×C324S4
C1C22C2×C6C62C32×A4C324S4 — C2×C324S4
C32×A4 — C2×C324S4
C1C2

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 2A2B2C2D2E3A3B3C3D3E···3M4A4B6A6B6C6D6E···6L6M···6U
order12222233333···34466666···66···6
size1133545422228···8545422226···68···8

42 irreducible representations

dim11122223366
type+++++++++++
imageC1C2C2S3S3D6D6S4C2×S4C3⋊S4C2×C3⋊S4
kernelC2×C324S4C324S4A4×C3×C6C6×A4C2×C62C3×A4C62C3×C6C32C6C3
# reps1211211212244

Matrix representation of C2×C324S4 in GL7(ℤ)

-1000000
0-100000
00-10000
000-1000
0000-100
00000-10
000000-1
,
0-100000
1-100000
00-11000
00-10000
0000-100
00000-10
0000001
,
0-100000
1-100000
000-1000
001-1000
0000-100
0000010
000000-1
,
0-100000
1-100000
0010000
0001000
0000001
0000100
0000010
,
1000000
1-100000
00-10000
00-11000
0000-100
000000-1
00000-10

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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