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G = D64S32order 432 = 24·33

1st semidirect product of D6 and S32 acting via S32/C3⋊S3=C2

metabelian, supersoluble, monomial

Aliases: D64S32, (S3×C6)⋊9D6, C337(C2×D4), C3⋊Dic39D6, D6⋊S32S3, C3215(S3×D4), C337D45C2, C34(Dic3⋊D6), (C32×C6).6C23, C2.6S33, (C2×S32)⋊3S3, (S32×C6)⋊3C2, C6.6(C2×S32), (C3×C3⋊S3)⋊2D4, C32(S3×C3⋊D4), (S3×C3×C6)⋊3C22, C339(C2×C4)⋊1C2, C3⋊S34(C3⋊D4), (C2×C3⋊S3).30D6, (C3×D6⋊S3)⋊3C2, C3210(C2×C3⋊D4), (C6×C3⋊S3).17C22, (C3×C6).55(C22×S3), (C3×C3⋊Dic3)⋊3C22, (C2×C33⋊C2)⋊2C22, (C2×S3×C3⋊S3)⋊2C2, SmallGroup(432,599)

Series: Derived Chief Lower central Upper central

C1C32×C6 — D64S32
C1C3C32C33C32×C6S3×C3×C6S32×C6 — D64S32
C33C32×C6 — D64S32
C1C2

Generators and relations for D64S32
 G = < a,b,c,d,e,f | a6=b2=c3=d2=e3=f2=1, bab=a-1, ac=ca, ad=da, ae=ea, af=fa, bc=cb, dbd=fbf=a3b, be=eb, dcd=c-1, ce=ec, cf=fc, de=ed, df=fd, fef=e-1 >

Subgroups: 2012 in 290 conjugacy classes, 50 normal (18 characteristic)
C1, C2, C2 [×6], C3, C3 [×2], C3 [×4], C4 [×2], C22 [×9], S3 [×18], C6, C6 [×2], C6 [×16], C2×C4, D4 [×4], C23 [×2], C32, C32 [×2], C32 [×4], Dic3 [×6], C12 [×2], D6, D6 [×2], D6 [×24], C2×C6 [×12], C2×D4, C3×S3 [×16], C3⋊S3 [×2], C3⋊S3 [×9], C3×C6, C3×C6 [×2], C3×C6 [×7], C4×S3 [×2], D12 [×2], C2×Dic3, C3⋊D4 [×10], C3×D4 [×2], C22×S3 [×6], C22×C6, C33, C3×Dic3 [×6], C3⋊Dic3 [×2], S32 [×10], S3×C6 [×6], S3×C6 [×10], C2×C3⋊S3, C2×C3⋊S3 [×9], C62 [×3], S3×D4 [×2], C2×C3⋊D4, S3×C32 [×3], C3×C3⋊S3 [×2], C33⋊C2, C32×C6, S3×Dic3 [×2], C6.D6, D6⋊S3 [×2], C3⋊D12 [×6], C3×C3⋊D4 [×4], C327D4 [×2], C2×S32, C2×S32 [×3], S3×C2×C6 [×2], C22×C3⋊S3, C3×C3⋊Dic3 [×2], C3×S32 [×2], S3×C3⋊S3 [×2], S3×C3×C6, S3×C3×C6 [×2], C6×C3⋊S3, C2×C33⋊C2, S3×C3⋊D4 [×2], Dic3⋊D6, C3×D6⋊S3 [×2], C337D4 [×2], C339(C2×C4), S32×C6, C2×S3×C3⋊S3, D64S32
Quotients: C1, C2 [×7], C22 [×7], S3 [×3], D4 [×2], C23, D6 [×9], C2×D4, C3⋊D4 [×2], C22×S3 [×3], S32 [×3], S3×D4 [×2], C2×C3⋊D4, C2×S32 [×3], S3×C3⋊D4 [×2], Dic3⋊D6, S33, D64S32

Permutation representations of D64S32
On 24 points - transitive group 24T1299
Generators in S24
(1 2 3 4 5 6)(7 8 9 10 11 12)(13 14 15 16 17 18)(19 20 21 22 23 24)
(1 12)(2 11)(3 10)(4 9)(5 8)(6 7)(13 19)(14 24)(15 23)(16 22)(17 21)(18 20)
(1 5 3)(2 6 4)(7 9 11)(8 10 12)(13 15 17)(14 16 18)(19 23 21)(20 24 22)
(1 17)(2 18)(3 13)(4 14)(5 15)(6 16)(7 19)(8 20)(9 21)(10 22)(11 23)(12 24)
(1 3 5)(2 4 6)(7 11 9)(8 12 10)(13 15 17)(14 16 18)(19 23 21)(20 24 22)
(1 19)(2 20)(3 21)(4 22)(5 23)(6 24)(7 17)(8 18)(9 13)(10 14)(11 15)(12 16)

G:=sub<Sym(24)| (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24), (1,12)(2,11)(3,10)(4,9)(5,8)(6,7)(13,19)(14,24)(15,23)(16,22)(17,21)(18,20), (1,5,3)(2,6,4)(7,9,11)(8,10,12)(13,15,17)(14,16,18)(19,23,21)(20,24,22), (1,17)(2,18)(3,13)(4,14)(5,15)(6,16)(7,19)(8,20)(9,21)(10,22)(11,23)(12,24), (1,3,5)(2,4,6)(7,11,9)(8,12,10)(13,15,17)(14,16,18)(19,23,21)(20,24,22), (1,19)(2,20)(3,21)(4,22)(5,23)(6,24)(7,17)(8,18)(9,13)(10,14)(11,15)(12,16)>;

G:=Group( (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24), (1,12)(2,11)(3,10)(4,9)(5,8)(6,7)(13,19)(14,24)(15,23)(16,22)(17,21)(18,20), (1,5,3)(2,6,4)(7,9,11)(8,10,12)(13,15,17)(14,16,18)(19,23,21)(20,24,22), (1,17)(2,18)(3,13)(4,14)(5,15)(6,16)(7,19)(8,20)(9,21)(10,22)(11,23)(12,24), (1,3,5)(2,4,6)(7,11,9)(8,12,10)(13,15,17)(14,16,18)(19,23,21)(20,24,22), (1,19)(2,20)(3,21)(4,22)(5,23)(6,24)(7,17)(8,18)(9,13)(10,14)(11,15)(12,16) );

G=PermutationGroup([(1,2,3,4,5,6),(7,8,9,10,11,12),(13,14,15,16,17,18),(19,20,21,22,23,24)], [(1,12),(2,11),(3,10),(4,9),(5,8),(6,7),(13,19),(14,24),(15,23),(16,22),(17,21),(18,20)], [(1,5,3),(2,6,4),(7,9,11),(8,10,12),(13,15,17),(14,16,18),(19,23,21),(20,24,22)], [(1,17),(2,18),(3,13),(4,14),(5,15),(6,16),(7,19),(8,20),(9,21),(10,22),(11,23),(12,24)], [(1,3,5),(2,4,6),(7,11,9),(8,12,10),(13,15,17),(14,16,18),(19,23,21),(20,24,22)], [(1,19),(2,20),(3,21),(4,22),(5,23),(6,24),(7,17),(8,18),(9,13),(10,14),(11,15),(12,16)])

G:=TransitiveGroup(24,1299);

42 conjugacy classes

class 1 2A2B2C2D2E2F2G3A3B3C3D3E3F3G4A4B6A6B6C6D6E6F6G6H6I6J6K6L···6U6V6W12A12B
order12222222333333344666666666666···6661212
size116669954222444818182224446666812···1218183636

42 irreducible representations

dim11111122222224444488
type++++++++++++++++++
imageC1C2C2C2C2C2S3S3D4D6D6D6C3⋊D4S32S3×D4C2×S32S3×C3⋊D4Dic3⋊D6S33D64S32
kernelD64S32C3×D6⋊S3C337D4C339(C2×C4)S32×C6C2×S3×C3⋊S3D6⋊S3C2×S32C3×C3⋊S3C3⋊Dic3S3×C6C2×C3⋊S3C3⋊S3D6C32C6C3C3C2C1
# reps12211121226143234211

Matrix representation of D64S32 in GL8(ℤ)

0-1000000
11000000
00110000
00-100000
00000-100
00001100
00000011
000000-10
,
00110000
00-100000
0-1000000
11000000
000000-1-1
00000010
00000100
0000-1-100
,
-1-1000000
10000000
00-1-10000
00100000
00000100
0000-1-100
00000001
000000-1-1
,
00001000
00000100
00000010
00000001
10000000
01000000
00100000
00010000
,
-1-1000000
10000000
00-1-10000
00100000
0000-1-100
00001000
000000-1-1
00000010
,
00000010
000000-1-1
00001000
0000-1-100
00100000
00-1-10000
10000000
-1-1000000

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

D64S32 in GAP, Magma, Sage, TeX

D_6\rtimes_4S_3^2
% in TeX

G:=Group("D6:4S3^2");
// GroupNames label

G:=SmallGroup(432,599);
// by ID

G=gap.SmallGroup(432,599);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,254,135,298,2028,14118]);
// Polycyclic

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

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