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G = D6⋊S32order 432 = 24·33

3rd semidirect product of D6 and S32 acting via S32/C3×S3=C2

metabelian, supersoluble, monomial

Aliases: D63S32, Dic33S32, C338(C2×D4), (S3×C6)⋊10D6, C3⋊D123S3, (C3×Dic3)⋊1D6, C3216(S3×D4), C336D42C2, C34(D6⋊D6), C335C44C22, (C32×C6).7C23, (C32×Dic3)⋊2C22, C2.7S33, (S32×C6)⋊4C2, (C2×S32)⋊4S3, C6.7(C2×S32), (C2×C3⋊S3)⋊9D6, (C3×C3⋊S3)⋊3D4, C33(S3×C3⋊D4), (S3×C3×C6)⋊4C22, C3⋊S33(C3⋊D4), (C6×C3⋊S3)⋊3C22, (Dic3×C3⋊S3)⋊1C2, C328(C2×C3⋊D4), (C3×C3⋊D12)⋊2C2, (C2×C324D6)⋊1C2, (C3×C6).56(C22×S3), SmallGroup(432,600)

Series: Derived Chief Lower central Upper central

C1C32×C6 — D6⋊S32
C1C3C32C33C32×C6S3×C3×C6S32×C6 — D6⋊S32
C33C32×C6 — D6⋊S32
C1C2

Generators and relations for D6⋊S32
 G = < a,b,c,d,e,f | a3=b3=c3=d2=e4=f2=1, ab=ba, ac=ca, ad=da, eae-1=a-1, af=fa, bc=cb, dbd=ebe-1=fbf=b-1, dcd=ece-1=c-1, cf=fc, de=ed, df=fd, fef=e-1 >

Subgroups: 1676 in 270 conjugacy classes, 50 normal (18 characteristic)
C1, C2, C2 [×6], C3, C3 [×2], C3 [×4], C4 [×2], C22 [×9], S3 [×14], C6, C6 [×2], C6 [×14], C2×C4, D4 [×4], C23 [×2], C32, C32 [×2], C32 [×4], Dic3, Dic3 [×7], C12 [×3], D6 [×2], D6 [×19], C2×C6 [×11], C2×D4, C3×S3 [×18], C3⋊S3 [×2], C3⋊S3 [×2], C3×C6, C3×C6 [×2], C3×C6 [×6], C4×S3 [×3], D12 [×2], C2×Dic3, C3⋊D4 [×10], C3×D4 [×2], C22×S3 [×5], C22×C6, C33, C3×Dic3 [×2], C3×Dic3, C3⋊Dic3 [×7], C3×C12, S32 [×8], S3×C6 [×4], S3×C6 [×15], C2×C3⋊S3, C2×C3⋊S3 [×2], C62 [×2], S3×D4 [×2], C2×C3⋊D4, S3×C32 [×2], C3×C3⋊S3 [×2], C3×C3⋊S3 [×2], C32×C6, S3×Dic3 [×3], D6⋊S3 [×6], C3⋊D12 [×2], C3×D12 [×2], C3×C3⋊D4 [×2], C4×C3⋊S3, C327D4 [×2], C2×S32, C2×S32 [×3], S3×C2×C6 [×2], C32×Dic3, C335C4, C3×S32 [×2], C324D6 [×2], S3×C3×C6 [×2], C6×C3⋊S3, C6×C3⋊S3 [×2], D6⋊D6, S3×C3⋊D4 [×2], C3×C3⋊D12 [×2], Dic3×C3⋊S3, C336D4 [×2], S32×C6, C2×C324D6, D6⋊S32
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], D6⋊D6, S3×C3⋊D4 [×2], S33, D6⋊S32

Smallest permutation representation of D6⋊S32
On 48 points
Generators in S48
(1 44 45)(2 46 41)(3 42 47)(4 48 43)(5 28 24)(6 21 25)(7 26 22)(8 23 27)(9 37 30)(10 31 38)(11 39 32)(12 29 40)(13 18 35)(14 36 19)(15 20 33)(16 34 17)
(1 44 45)(2 46 41)(3 42 47)(4 48 43)(5 24 28)(6 25 21)(7 22 26)(8 27 23)(9 30 37)(10 38 31)(11 32 39)(12 40 29)(13 18 35)(14 36 19)(15 20 33)(16 34 17)
(1 45 44)(2 41 46)(3 47 42)(4 43 48)(5 28 24)(6 21 25)(7 26 22)(8 23 27)(9 30 37)(10 38 31)(11 32 39)(12 40 29)(13 18 35)(14 36 19)(15 20 33)(16 34 17)
(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 44)(22 41)(23 42)(24 43)(25 45)(26 46)(27 47)(28 48)(29 35)(30 36)(31 33)(32 34)
(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 30)(2 29)(3 32)(4 31)(5 33)(6 36)(7 35)(8 34)(9 44)(10 43)(11 42)(12 41)(13 26)(14 25)(15 28)(16 27)(17 23)(18 22)(19 21)(20 24)(37 45)(38 48)(39 47)(40 46)

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

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

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

42 conjugacy classes

class 1 2A2B2C2D2E2F2G3A3B3C3D3E3F3G4A4B6A6B6C6D6E6F6G6H6I6J6K6L···6Q6R6S6T6U12A12B12C12D
order12222222333333344666666666666···6666612121212
size116699181822244486542224446666812···121818363612121212

42 irreducible representations

dim111111222222244444488
type+++++++++++++++++-
imageC1C2C2C2C2C2S3S3D4D6D6D6C3⋊D4S32S32S3×D4C2×S32D6⋊D6S3×C3⋊D4S33D6⋊S32
kernelD6⋊S32C3×C3⋊D12Dic3×C3⋊S3C336D4S32×C6C2×C324D6C3⋊D12C2×S32C3×C3⋊S3C3×Dic3S3×C6C2×C3⋊S3C3⋊S3Dic3D6C32C6C3C3C2C1
# reps121211212243412232411

Matrix representation of D6⋊S32 in GL8(ℤ)

10000000
01000000
00100000
00010000
00001000
00000100
0000000-1
0000001-1
,
10000000
01000000
00100000
00010000
0000-1-100
00001000
00000010
00000001
,
10000000
01000000
000-10000
001-10000
00001000
00000100
00000010
00000001
,
10000000
01000000
000-10000
00-100000
0000-1000
00001100
00000010
00000001
,
01000000
-10000000
00010000
00100000
00001000
0000-1-100
0000000-1
000000-10
,
-10000000
01000000
00-100000
000-10000
00001000
0000-1-100
000000-10
0000000-1

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

D6⋊S32 in GAP, Magma, Sage, TeX

D_6\rtimes S_3^2
% in TeX

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

G:=SmallGroup(432,600);
// by ID

G=gap.SmallGroup(432,600);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,254,135,58,298,2028,14118]);
// Polycyclic

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

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