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G = Q8×C32⋊C6  order 432 = 24·33

Direct product of Q8 and C32⋊C6

direct product, metabelian, supersoluble, monomial

Aliases: Q8×C32⋊C6, He3⋊7(C2×Q8), (Q8×He3)⋊3C2, C32⋊4(S3×Q8), C32⋊3(C6×Q8), C12.21(S3×C6), He3⋊3Q8⋊9C2, (C3×C12).28D6, (Q8×C32)⋊3C6, (Q8×C32)⋊5S3, C32⋊4Q8⋊4C6, (C4×He3).22C22, (C2×He3).25C23, C32⋊C12.13C22, C3.2(C3×S3×Q8), C6.39(S3×C2×C6), C3⋊S3⋊2(C3×Q8), (Q8×C3⋊S3)⋊2C3, (C4×C3⋊S3).1C6, (C3×C12).6(C2×C6), C4.6(C2×C32⋊C6), (C4×C32⋊C6).3C2, C3⋊Dic3.4(C2×C6), (C3×C6).7(C22×C6), (C3×Q8).29(C3×S3), (C3×C6).31(C22×S3), C2.8(C22×C32⋊C6), (C2×C32⋊C6).14C22, (C2×C3⋊S3).12(C2×C6), SmallGroup(432,368)

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, C32⋊4Q8, C4×C3⋊S3, Q8×C32, Q8×C32, C32⋊C12, C4×He3, C2×C32⋊C6, C3×S3×Q8, Q8×C3⋊S3, He3⋊3Q8, 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 2A2B2C3A3B3C3D3E3F4A4B4C4D4E4F6A6B6C6D6E6F6G6H6I6J12A12B12C12D···12I12J···12R12S···12X
order1222333333444444666666666612121212···1212···1212···12
size119923366622218181823366699994446···612···1218···18

50 irreducible representations

dim11111111122222224466
type++++-+-+-++
imageC1C2C2C2C3C6C6C6Q8×C32⋊C6S3Q8D6C3×S3C3×Q8S3×C6S3×Q8C3×S3×Q8C32⋊C6C2×C32⋊C6
kernelQ8×C32⋊C6He3⋊3Q8C4×C32⋊C6Q8×He3Q8×C3⋊S3C32⋊4Q8C4×C3⋊S3Q8×C32C1Q8×C32C32⋊C6C3×C12C3×Q8C3⋊S3C12C32C3Q8C4
# reps1331266211232461213

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

0010000000
0001000000
12000000000
01200000000
00001200000
00000120000
00000012000
00000001200
00000000120
00000000012
,
10090000000
01009000000
9030000000
0903000000
00001200000
00000120000
00000012000
00000001200
00000000120
00000000012
,
01200000000
11200000000
00012000000
00112000000
00001210000
00001200000
00000001200
00000011200
0000100110
0000100101
,
1000000000
0100000000
0010000000
0001000000
00000120000
00001120000
00000001200
00000011200
00001201201212
0000010110
,
01000000000
10000000000
00010000000
00100000000
0000000100
0000001000
0000121212121112
00000000121
0000000010
0000010010

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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