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## G = C2×C4×C9⋊C6order 432 = 24·33

### Direct product of C2×C4 and C9⋊C6

Series: Derived Chief Lower central Upper central

 Derived series C1 — C9 — C2×C4×C9⋊C6
 Chief series C1 — C3 — C9 — C18 — C2×3- 1+2 — C2×C9⋊C6 — C22×C9⋊C6 — C2×C4×C9⋊C6
 Lower central C9 — C2×C4×C9⋊C6
 Upper central C1 — C2×C4

Generators and relations for C2×C4×C9⋊C6
G = < a,b,c,d | a2=b4=c9=d6=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c2 >

Subgroups: 574 in 170 conjugacy classes, 78 normal (26 characteristic)
C1, C2, C2, C2, C3, C3, C4, C4, C22, C22, S3, C6, C6, C6, C2×C4, C2×C4, C23, C9, C9, C32, Dic3, C12, C12, D6, C2×C6, C2×C6, C22×C4, D9, C18, C18, C18, C3×S3, C3×C6, C3×C6, C4×S3, C2×Dic3, C2×C12, C2×C12, C22×S3, C22×C6, 3- 1+2, Dic9, C36, C36, D18, C2×C18, C2×C18, C3×Dic3, C3×C12, S3×C6, C62, S3×C2×C4, C22×C12, C9⋊C6, C2×3- 1+2, C2×3- 1+2, C4×D9, C2×Dic9, C2×C36, C2×C36, C22×D9, S3×C12, C6×Dic3, C6×C12, S3×C2×C6, C9⋊C12, C4×3- 1+2, C2×C9⋊C6, C22×3- 1+2, C2×C4×D9, S3×C2×C12, C4×C9⋊C6, C2×C9⋊C12, C2×C4×3- 1+2, C22×C9⋊C6, C2×C4×C9⋊C6
Quotients: C1, C2, C3, C4, C22, S3, C6, C2×C4, C23, C12, D6, C2×C6, C22×C4, C3×S3, C4×S3, C2×C12, C22×S3, C22×C6, S3×C6, S3×C2×C4, C22×C12, C9⋊C6, S3×C12, S3×C2×C6, C2×C9⋊C6, S3×C2×C12, C4×C9⋊C6, C22×C9⋊C6, C2×C4×C9⋊C6

Smallest permutation representation of C2×C4×C9⋊C6
On 72 points
Generators in S72
(1 38)(2 39)(3 40)(4 41)(5 42)(6 43)(7 44)(8 45)(9 37)(10 46)(11 47)(12 48)(13 49)(14 50)(15 51)(16 52)(17 53)(18 54)(19 55)(20 56)(21 57)(22 58)(23 59)(24 60)(25 61)(26 62)(27 63)(28 64)(29 65)(30 66)(31 67)(32 68)(33 69)(34 70)(35 71)(36 72)
(1 29 11 20)(2 30 12 21)(3 31 13 22)(4 32 14 23)(5 33 15 24)(6 34 16 25)(7 35 17 26)(8 36 18 27)(9 28 10 19)(37 64 46 55)(38 65 47 56)(39 66 48 57)(40 67 49 58)(41 68 50 59)(42 69 51 60)(43 70 52 61)(44 71 53 62)(45 72 54 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)
(1 38)(2 43 8 37 5 40)(3 39 6 45 9 42)(4 44)(7 41)(10 51 13 48 16 54)(11 47)(12 52 18 46 15 49)(14 53)(17 50)(19 60 22 57 25 63)(20 56)(21 61 27 55 24 58)(23 62)(26 59)(28 69 31 66 34 72)(29 65)(30 70 36 64 33 67)(32 71)(35 68)

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

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

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

80 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 3C 4A 4B 4C 4D 4E 4F 4G 4H 6A 6B 6C 6D ··· 6I 6J ··· 6Q 9A 9B 9C 12A 12B 12C 12D 12E ··· 12L 12M ··· 12T 18A ··· 18I 36A ··· 36L order 1 2 2 2 2 2 2 2 3 3 3 4 4 4 4 4 4 4 4 6 6 6 6 ··· 6 6 ··· 6 9 9 9 12 12 12 12 12 ··· 12 12 ··· 12 18 ··· 18 36 ··· 36 size 1 1 1 1 9 9 9 9 2 3 3 1 1 1 1 9 9 9 9 2 2 2 3 ··· 3 9 ··· 9 6 6 6 2 2 2 2 3 ··· 3 9 ··· 9 6 ··· 6 6 ··· 6

80 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 6 6 6 6 type + + + + + + + + + + + image C1 C2 C2 C2 C2 C3 C4 C6 C6 C6 C6 C12 S3 D6 D6 C3×S3 C4×S3 S3×C6 S3×C6 S3×C12 C9⋊C6 C2×C9⋊C6 C2×C9⋊C6 C4×C9⋊C6 kernel C2×C4×C9⋊C6 C4×C9⋊C6 C2×C9⋊C12 C2×C4×3- 1+2 C22×C9⋊C6 C2×C4×D9 C2×C9⋊C6 C4×D9 C2×Dic9 C2×C36 C22×D9 D18 C6×C12 C3×C12 C62 C2×C12 C3×C6 C12 C2×C6 C6 C2×C4 C4 C22 C2 # reps 1 4 1 1 1 2 8 8 2 2 2 16 1 2 1 2 4 4 2 8 1 2 1 4

Matrix representation of C2×C4×C9⋊C6 in GL8(𝔽37)

 36 0 0 0 0 0 0 0 0 36 0 0 0 0 0 0 0 0 36 0 0 0 0 0 0 0 0 36 0 0 0 0 0 0 0 0 36 0 0 0 0 0 0 0 0 36 0 0 0 0 0 0 0 0 36 0 0 0 0 0 0 0 0 36
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 31 0 0 0 0 0 0 0 0 31 0 0 0 0 0 0 0 0 31 0 0 0 0 0 0 0 0 31 0 0 0 0 0 0 0 0 31 0 0 0 0 0 0 0 0 31
,
 0 1 0 0 0 0 0 0 36 36 0 0 0 0 0 0 0 0 0 0 36 1 0 0 0 0 0 0 36 0 0 0 0 0 36 36 36 36 36 35 0 0 0 0 0 0 1 36 0 0 0 0 1 0 0 1 0 0 1 0 1 0 0 1
,
 11 0 0 0 0 0 0 0 26 26 0 0 0 0 0 0 0 0 0 36 0 0 0 0 0 0 36 0 0 0 0 0 0 0 1 1 1 1 2 1 0 0 0 0 0 0 1 36 0 0 0 0 0 36 36 0 0 0 0 0 36 0 36 0

G:=sub<GL(8,GF(37))| [36,0,0,0,0,0,0,0,0,36,0,0,0,0,0,0,0,0,36,0,0,0,0,0,0,0,0,36,0,0,0,0,0,0,0,0,36,0,0,0,0,0,0,0,0,36,0,0,0,0,0,0,0,0,36,0,0,0,0,0,0,0,0,36],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,31,0,0,0,0,0,0,0,0,31,0,0,0,0,0,0,0,0,31,0,0,0,0,0,0,0,0,31,0,0,0,0,0,0,0,0,31,0,0,0,0,0,0,0,0,31],[0,36,0,0,0,0,0,0,1,36,0,0,0,0,0,0,0,0,0,0,36,0,0,1,0,0,0,0,36,0,0,0,0,0,36,36,36,0,1,1,0,0,1,0,36,0,0,0,0,0,0,0,36,1,0,0,0,0,0,0,35,36,1,1],[11,26,0,0,0,0,0,0,0,26,0,0,0,0,0,0,0,0,0,36,1,0,0,0,0,0,36,0,1,0,0,0,0,0,0,0,1,0,0,36,0,0,0,0,1,0,36,0,0,0,0,0,2,1,36,36,0,0,0,0,1,36,0,0] >;

C2×C4×C9⋊C6 in GAP, Magma, Sage, TeX

C_2\times C_4\times C_9\rtimes C_6
% in TeX

G:=Group("C2xC4xC9:C6");
// GroupNames label

G:=SmallGroup(432,353);
// by ID

G=gap.SmallGroup(432,353);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-3,-3,142,10085,1034,292,14118]);
// Polycyclic

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

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