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## G = C6×C9⋊D4order 432 = 24·33

### Direct product of C6 and C9⋊D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C18 — C6×C9⋊D4
 Chief series C1 — C3 — C9 — C18 — C3×C18 — C6×D9 — C2×C6×D9 — C6×C9⋊D4
 Lower central C9 — C18 — C6×C9⋊D4
 Upper central C1 — C2×C6 — C22×C6

Generators and relations for C6×C9⋊D4
G = < a,b,c,d | a6=b9=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=c-1 >

Subgroups: 654 in 194 conjugacy classes, 70 normal (30 characteristic)
C1, C2, C2, C2, C3, C3, C4, C22, C22, C22, S3, C6, C6, C6, C2×C4, D4, C23, C23, C9, C9, C32, Dic3, C12, D6, C2×C6, C2×C6, C2×C6, C2×D4, D9, C18, C18, C18, C3×S3, C3×C6, C3×C6, C3×C6, C2×Dic3, C3⋊D4, C2×C12, C3×D4, C22×S3, C22×C6, C22×C6, C3×C9, Dic9, D18, D18, C2×C18, C2×C18, C2×C18, C3×Dic3, S3×C6, C62, C62, C62, C2×C3⋊D4, C6×D4, C3×D9, C3×C18, C3×C18, C3×C18, C2×Dic9, C9⋊D4, C22×D9, C22×C18, C22×C18, C6×Dic3, C3×C3⋊D4, S3×C2×C6, C2×C62, C3×Dic9, C6×D9, C6×D9, C6×C18, C6×C18, C6×C18, C2×C9⋊D4, C6×C3⋊D4, C6×Dic9, C3×C9⋊D4, C2×C6×D9, C2×C6×C18, C6×C9⋊D4
Quotients: C1, C2, C3, C22, S3, C6, D4, C23, D6, C2×C6, C2×D4, D9, C3×S3, C3⋊D4, C3×D4, C22×S3, C22×C6, D18, S3×C6, C2×C3⋊D4, C6×D4, C3×D9, C9⋊D4, C22×D9, C3×C3⋊D4, S3×C2×C6, C6×D9, C2×C9⋊D4, C6×C3⋊D4, C3×C9⋊D4, C2×C6×D9, C6×C9⋊D4

Smallest permutation representation of C6×C9⋊D4
On 72 points
Generators in S72
(1 25 7 22 4 19)(2 26 8 23 5 20)(3 27 9 24 6 21)(10 31 16 28 13 34)(11 32 17 29 14 35)(12 33 18 30 15 36)(37 61 40 55 43 58)(38 62 41 56 44 59)(39 63 42 57 45 60)(46 70 49 64 52 67)(47 71 50 65 53 68)(48 72 51 66 54 69)
(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)
(1 50 13 41)(2 49 14 40)(3 48 15 39)(4 47 16 38)(5 46 17 37)(6 54 18 45)(7 53 10 44)(8 52 11 43)(9 51 12 42)(19 71 28 62)(20 70 29 61)(21 69 30 60)(22 68 31 59)(23 67 32 58)(24 66 33 57)(25 65 34 56)(26 64 35 55)(27 72 36 63)
(1 59)(2 58)(3 57)(4 56)(5 55)(6 63)(7 62)(8 61)(9 60)(10 71)(11 70)(12 69)(13 68)(14 67)(15 66)(16 65)(17 64)(18 72)(19 44)(20 43)(21 42)(22 41)(23 40)(24 39)(25 38)(26 37)(27 45)(28 53)(29 52)(30 51)(31 50)(32 49)(33 48)(34 47)(35 46)(36 54)

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

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

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

126 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 3C 3D 3E 4A 4B 6A ··· 6F 6G ··· 6AE 6AF 6AG 6AH 6AI 9A ··· 9I 12A 12B 12C 12D 18A ··· 18BK order 1 2 2 2 2 2 2 2 3 3 3 3 3 4 4 6 ··· 6 6 ··· 6 6 6 6 6 9 ··· 9 12 12 12 12 18 ··· 18 size 1 1 1 1 2 2 18 18 1 1 2 2 2 18 18 1 ··· 1 2 ··· 2 18 18 18 18 2 ··· 2 18 18 18 18 2 ··· 2

126 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + + + + + + + image C1 C2 C2 C2 C2 C3 C6 C6 C6 C6 S3 D4 D6 D9 C3×S3 C3×D4 C3⋊D4 D18 S3×C6 C3×D9 C9⋊D4 C3×C3⋊D4 C6×D9 C3×C9⋊D4 kernel C6×C9⋊D4 C6×Dic9 C3×C9⋊D4 C2×C6×D9 C2×C6×C18 C2×C9⋊D4 C2×Dic9 C9⋊D4 C22×D9 C22×C18 C2×C62 C3×C18 C62 C22×C6 C22×C6 C18 C3×C6 C2×C6 C2×C6 C23 C6 C6 C22 C2 # reps 1 1 4 1 1 2 2 8 2 2 1 2 3 3 2 4 4 9 6 6 12 8 18 24

Matrix representation of C6×C9⋊D4 in GL4(𝔽37) generated by

 11 0 0 0 0 11 0 0 0 0 36 0 0 0 0 36
,
 12 0 0 0 0 34 0 0 0 0 26 0 0 0 2 10
,
 0 1 0 0 1 0 0 0 0 0 36 8 0 0 9 1
,
 0 36 0 0 36 0 0 0 0 0 1 29 0 0 0 36
G:=sub<GL(4,GF(37))| [11,0,0,0,0,11,0,0,0,0,36,0,0,0,0,36],[12,0,0,0,0,34,0,0,0,0,26,2,0,0,0,10],[0,1,0,0,1,0,0,0,0,0,36,9,0,0,8,1],[0,36,0,0,36,0,0,0,0,0,1,0,0,0,29,36] >;

C6×C9⋊D4 in GAP, Magma, Sage, TeX

C_6\times C_9\rtimes D_4
% in TeX

G:=Group("C6xC9:D4");
// GroupNames label

G:=SmallGroup(432,374);
// by ID

G=gap.SmallGroup(432,374);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-3,-3,590,10085,292,14118]);
// Polycyclic

G:=Group<a,b,c,d|a^6=b^9=c^4=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations

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