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## G = Q8×C32⋊C6order 432 = 24·33

### Direct product of Q8 and C32⋊C6

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6 — Q8×C32⋊C6
 Chief series C1 — C3 — C32 — C3×C6 — C2×He3 — C2×C32⋊C6 — C4×C32⋊C6 — Q8×C32⋊C6
 Lower central C32 — C3×C6 — Q8×C32⋊C6
 Upper central C1 — C2 — Q8

Generators and relations for Q8×C32⋊C6
G = < a,b,c,d,e | a4=c3=d3=e6=1, b2=a2, bab-1=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece-1=c-1d-1, ede-1=d-1 >

Subgroups: 569 in 145 conjugacy classes, 56 normal (19 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, C6, C6, C2×C4, Q8, Q8, C32, C32, Dic3, C12, C12, D6, C2×C6, C2×Q8, C3×S3, C3⋊S3, C3×C6, C3×C6, Dic6, C4×S3, C2×C12, C3×Q8, C3×Q8, He3, C3×Dic3, C3⋊Dic3, C3×C12, C3×C12, S3×C6, C2×C3⋊S3, S3×Q8, C6×Q8, C32⋊C6, C2×He3, C3×Dic6, S3×C12, C324Q8, C4×C3⋊S3, Q8×C32, Q8×C32, C32⋊C12, C4×He3, C2×C32⋊C6, C3×S3×Q8, Q8×C3⋊S3, He33Q8, C4×C32⋊C6, Q8×He3, Q8×C32⋊C6
Quotients: C1, C2, C3, C22, S3, C6, Q8, C23, D6, C2×C6, C2×Q8, C3×S3, C3×Q8, C22×S3, C22×C6, S3×C6, S3×Q8, C6×Q8, C32⋊C6, S3×C2×C6, C2×C32⋊C6, C3×S3×Q8, C22×C32⋊C6, Q8×C32⋊C6

Smallest permutation representation of Q8×C32⋊C6
On 72 points
Generators in S72
(1 40 4 37)(2 41 5 38)(3 42 6 39)(7 57 10 60)(8 58 11 55)(9 59 12 56)(13 53 71 65)(14 54 72 66)(15 49 67 61)(16 50 68 62)(17 51 69 63)(18 52 70 64)(19 48 30 36)(20 43 25 31)(21 44 26 32)(22 45 27 33)(23 46 28 34)(24 47 29 35)
(1 58 4 55)(2 59 5 56)(3 60 6 57)(7 42 10 39)(8 37 11 40)(9 38 12 41)(13 47 71 35)(14 48 72 36)(15 43 67 31)(16 44 68 32)(17 45 69 33)(18 46 70 34)(19 66 30 54)(20 61 25 49)(21 62 26 50)(22 63 27 51)(23 64 28 52)(24 65 29 53)
(1 23 25)(3 27 19)(4 28 20)(6 22 30)(7 69 14)(8 70 15)(10 17 72)(11 18 67)(31 40 46)(33 48 42)(34 43 37)(36 39 45)(49 58 64)(51 66 60)(52 61 55)(54 57 63)
(1 25 23)(2 24 26)(3 27 19)(4 20 28)(5 29 21)(6 22 30)(7 69 14)(8 15 70)(9 71 16)(10 17 72)(11 67 18)(12 13 68)(31 46 40)(32 41 47)(33 48 42)(34 37 43)(35 44 38)(36 39 45)(49 64 58)(50 59 65)(51 66 60)(52 55 61)(53 62 56)(54 57 63)
(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)(55 56 57 58 59 60)(61 62 63 64 65 66)(67 68 69 70 71 72)

G:=sub<Sym(72)| (1,40,4,37)(2,41,5,38)(3,42,6,39)(7,57,10,60)(8,58,11,55)(9,59,12,56)(13,53,71,65)(14,54,72,66)(15,49,67,61)(16,50,68,62)(17,51,69,63)(18,52,70,64)(19,48,30,36)(20,43,25,31)(21,44,26,32)(22,45,27,33)(23,46,28,34)(24,47,29,35), (1,58,4,55)(2,59,5,56)(3,60,6,57)(7,42,10,39)(8,37,11,40)(9,38,12,41)(13,47,71,35)(14,48,72,36)(15,43,67,31)(16,44,68,32)(17,45,69,33)(18,46,70,34)(19,66,30,54)(20,61,25,49)(21,62,26,50)(22,63,27,51)(23,64,28,52)(24,65,29,53), (1,23,25)(3,27,19)(4,28,20)(6,22,30)(7,69,14)(8,70,15)(10,17,72)(11,18,67)(31,40,46)(33,48,42)(34,43,37)(36,39,45)(49,58,64)(51,66,60)(52,61,55)(54,57,63), (1,25,23)(2,24,26)(3,27,19)(4,20,28)(5,29,21)(6,22,30)(7,69,14)(8,15,70)(9,71,16)(10,17,72)(11,67,18)(12,13,68)(31,46,40)(32,41,47)(33,48,42)(34,37,43)(35,44,38)(36,39,45)(49,64,58)(50,59,65)(51,66,60)(52,55,61)(53,62,56)(54,57,63), (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)(55,56,57,58,59,60)(61,62,63,64,65,66)(67,68,69,70,71,72)>;

G:=Group( (1,40,4,37)(2,41,5,38)(3,42,6,39)(7,57,10,60)(8,58,11,55)(9,59,12,56)(13,53,71,65)(14,54,72,66)(15,49,67,61)(16,50,68,62)(17,51,69,63)(18,52,70,64)(19,48,30,36)(20,43,25,31)(21,44,26,32)(22,45,27,33)(23,46,28,34)(24,47,29,35), (1,58,4,55)(2,59,5,56)(3,60,6,57)(7,42,10,39)(8,37,11,40)(9,38,12,41)(13,47,71,35)(14,48,72,36)(15,43,67,31)(16,44,68,32)(17,45,69,33)(18,46,70,34)(19,66,30,54)(20,61,25,49)(21,62,26,50)(22,63,27,51)(23,64,28,52)(24,65,29,53), (1,23,25)(3,27,19)(4,28,20)(6,22,30)(7,69,14)(8,70,15)(10,17,72)(11,18,67)(31,40,46)(33,48,42)(34,43,37)(36,39,45)(49,58,64)(51,66,60)(52,61,55)(54,57,63), (1,25,23)(2,24,26)(3,27,19)(4,20,28)(5,29,21)(6,22,30)(7,69,14)(8,15,70)(9,71,16)(10,17,72)(11,67,18)(12,13,68)(31,46,40)(32,41,47)(33,48,42)(34,37,43)(35,44,38)(36,39,45)(49,64,58)(50,59,65)(51,66,60)(52,55,61)(53,62,56)(54,57,63), (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)(55,56,57,58,59,60)(61,62,63,64,65,66)(67,68,69,70,71,72) );

G=PermutationGroup([[(1,40,4,37),(2,41,5,38),(3,42,6,39),(7,57,10,60),(8,58,11,55),(9,59,12,56),(13,53,71,65),(14,54,72,66),(15,49,67,61),(16,50,68,62),(17,51,69,63),(18,52,70,64),(19,48,30,36),(20,43,25,31),(21,44,26,32),(22,45,27,33),(23,46,28,34),(24,47,29,35)], [(1,58,4,55),(2,59,5,56),(3,60,6,57),(7,42,10,39),(8,37,11,40),(9,38,12,41),(13,47,71,35),(14,48,72,36),(15,43,67,31),(16,44,68,32),(17,45,69,33),(18,46,70,34),(19,66,30,54),(20,61,25,49),(21,62,26,50),(22,63,27,51),(23,64,28,52),(24,65,29,53)], [(1,23,25),(3,27,19),(4,28,20),(6,22,30),(7,69,14),(8,70,15),(10,17,72),(11,18,67),(31,40,46),(33,48,42),(34,43,37),(36,39,45),(49,58,64),(51,66,60),(52,61,55),(54,57,63)], [(1,25,23),(2,24,26),(3,27,19),(4,20,28),(5,29,21),(6,22,30),(7,69,14),(8,15,70),(9,71,16),(10,17,72),(11,67,18),(12,13,68),(31,46,40),(32,41,47),(33,48,42),(34,37,43),(35,44,38),(36,39,45),(49,64,58),(50,59,65),(51,66,60),(52,55,61),(53,62,56),(54,57,63)], [(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),(55,56,57,58,59,60),(61,62,63,64,65,66),(67,68,69,70,71,72)]])

50 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 3D 3E 3F 4A 4B 4C 4D 4E 4F 6A 6B 6C 6D 6E 6F 6G 6H 6I 6J 12A 12B 12C 12D ··· 12I 12J ··· 12R 12S ··· 12X order 1 2 2 2 3 3 3 3 3 3 4 4 4 4 4 4 6 6 6 6 6 6 6 6 6 6 12 12 12 12 ··· 12 12 ··· 12 12 ··· 12 size 1 1 9 9 2 3 3 6 6 6 2 2 2 18 18 18 2 3 3 6 6 6 9 9 9 9 4 4 4 6 ··· 6 12 ··· 12 18 ··· 18

50 irreducible representations

 dim 1 1 1 1 1 1 1 1 12 2 2 2 2 2 2 4 4 6 6 type + + + + - + - + - + + image C1 C2 C2 C2 C3 C6 C6 C6 Q8×C32⋊C6 S3 Q8 D6 C3×S3 C3×Q8 S3×C6 S3×Q8 C3×S3×Q8 C32⋊C6 C2×C32⋊C6 kernel Q8×C32⋊C6 He3⋊3Q8 C4×C32⋊C6 Q8×He3 Q8×C3⋊S3 C32⋊4Q8 C4×C3⋊S3 Q8×C32 C1 Q8×C32 C32⋊C6 C3×C12 C3×Q8 C3⋊S3 C12 C32 C3 Q8 C4 # reps 1 3 3 1 2 6 6 2 1 1 2 3 2 4 6 1 2 1 3

Matrix representation of Q8×C32⋊C6 in GL10(𝔽13)

 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 0 12
,
 10 0 9 0 0 0 0 0 0 0 0 10 0 9 0 0 0 0 0 0 9 0 3 0 0 0 0 0 0 0 0 9 0 3 0 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 0 12
,
 0 12 0 0 0 0 0 0 0 0 1 12 0 0 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 1 12 0 0 0 0 0 0 0 0 0 0 12 1 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 1 12 0 0 0 0 0 0 1 0 0 1 1 0 0 0 0 0 1 0 0 1 0 1
,
 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 1 12 0 0 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 1 12 0 0 0 0 0 0 12 0 12 0 12 12 0 0 0 0 0 1 0 1 1 0
,
 0 10 0 0 0 0 0 0 0 0 10 0 0 0 0 0 0 0 0 0 0 0 0 10 0 0 0 0 0 0 0 0 10 0 0 0 0 0 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 12 12 12 12 11 12 0 0 0 0 0 0 0 0 12 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 1 0

G:=sub<GL(10,GF(13))| [0,0,12,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,12],[10,0,9,0,0,0,0,0,0,0,0,10,0,9,0,0,0,0,0,0,9,0,3,0,0,0,0,0,0,0,0,9,0,3,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,12],[0,1,0,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,0,0,0,0,12,12,0,0,1,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,12,1,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,12,0,0,0,0,0,12,12,0,0,0,1,0,0,0,0,0,0,0,1,12,0,0,0,0,0,0,0,12,12,0,1,0,0,0,0,0,0,0,0,12,1,0,0,0,0,0,0,0,0,12,0],[0,10,0,0,0,0,0,0,0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,10,0,0,0,0,0,0,0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,12,0,0,1,0,0,0,0,0,1,12,0,0,0,0,0,0,0,1,0,12,0,0,0,0,0,0,0,0,0,11,12,1,1,0,0,0,0,0,0,12,1,0,0] >;

Q8×C32⋊C6 in GAP, Magma, Sage, TeX

Q_8\times C_3^2\rtimes C_6
% in TeX

G:=Group("Q8xC3^2:C6");
// GroupNames label

G:=SmallGroup(432,368);
// by ID

G=gap.SmallGroup(432,368);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-3,-3,176,303,142,4037,1034,14118]);
// Polycyclic

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

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