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## G = C8×C9⋊C6order 432 = 24·33

### Direct product of C8 and C9⋊C6

Series: Derived Chief Lower central Upper central

 Derived series C1 — C9 — C8×C9⋊C6
 Chief series C1 — C3 — C9 — C18 — C36 — C4×3- 1+2 — C4×C9⋊C6 — C8×C9⋊C6
 Lower central C9 — C8×C9⋊C6
 Upper central C1 — C8

Generators and relations for C8×C9⋊C6
G = < a,b,c | a8=b9=c6=1, ab=ba, ac=ca, cbc-1=b2 >

Subgroups: 222 in 70 conjugacy classes, 34 normal (30 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, C6, C6, C8, C8, C2×C4, C9, C9, C32, Dic3, C12, C12, D6, C2×C6, C2×C8, D9, C18, C18, C3×S3, C3×C6, C3⋊C8, C24, C24, C4×S3, C2×C12, 3- 1+2, Dic9, C36, C36, D18, C3×Dic3, C3×C12, S3×C6, S3×C8, C2×C24, C9⋊C6, C2×3- 1+2, C9⋊C8, C72, C72, C4×D9, C3×C3⋊C8, C3×C24, S3×C12, C9⋊C12, C4×3- 1+2, C2×C9⋊C6, C8×D9, S3×C24, C9⋊C24, C8×3- 1+2, C4×C9⋊C6, C8×C9⋊C6
Quotients: C1, C2, C3, C4, C22, S3, C6, C8, C2×C4, C12, D6, C2×C6, C2×C8, C3×S3, C24, C4×S3, C2×C12, S3×C6, S3×C8, C2×C24, C9⋊C6, S3×C12, C2×C9⋊C6, S3×C24, C4×C9⋊C6, C8×C9⋊C6

Smallest permutation representation of C8×C9⋊C6
On 72 points
Generators in S72
(1 65 29 47 11 56 20 38)(2 66 30 48 12 57 21 39)(3 67 31 49 13 58 22 40)(4 68 32 50 14 59 23 41)(5 69 33 51 15 60 24 42)(6 70 34 52 16 61 25 43)(7 71 35 53 17 62 26 44)(8 72 36 54 18 63 27 45)(9 64 28 46 10 55 19 37)
(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)
(2 6 8 9 5 3)(4 7)(10 15 13 12 16 18)(14 17)(19 24 22 21 25 27)(23 26)(28 33 31 30 34 36)(32 35)(37 42 40 39 43 45)(41 44)(46 51 49 48 52 54)(50 53)(55 60 58 57 61 63)(59 62)(64 69 67 66 70 72)(68 71)

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

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

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

80 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 4A 4B 4C 4D 6A 6B 6C 6D 6E 6F 6G 8A 8B 8C 8D 8E 8F 8G 8H 9A 9B 9C 12A 12B 12C 12D 12E 12F 12G 12H 12I 12J 18A 18B 18C 24A 24B 24C 24D 24E ··· 24L 24M ··· 24T 36A ··· 36F 72A ··· 72L order 1 2 2 2 3 3 3 4 4 4 4 6 6 6 6 6 6 6 8 8 8 8 8 8 8 8 9 9 9 12 12 12 12 12 12 12 12 12 12 18 18 18 24 24 24 24 24 ··· 24 24 ··· 24 36 ··· 36 72 ··· 72 size 1 1 9 9 2 3 3 1 1 9 9 2 3 3 9 9 9 9 1 1 1 1 9 9 9 9 6 6 6 2 2 3 3 3 3 9 9 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 1 1 2 2 2 2 2 2 2 2 6 6 6 6 type + + + + + + + + image C1 C2 C2 C2 C3 C4 C4 C6 C6 C6 C8 C12 C12 C24 S3 D6 C3×S3 C4×S3 S3×C6 S3×C8 S3×C12 S3×C24 C9⋊C6 C2×C9⋊C6 C4×C9⋊C6 C8×C9⋊C6 kernel C8×C9⋊C6 C9⋊C24 C8×3- 1+2 C4×C9⋊C6 C8×D9 C9⋊C12 C2×C9⋊C6 C9⋊C8 C72 C4×D9 C9⋊C6 Dic9 D18 D9 C3×C24 C3×C12 C24 C3×C6 C12 C32 C6 C3 C8 C4 C2 C1 # reps 1 1 1 1 2 2 2 2 2 2 8 4 4 16 1 1 2 2 2 4 4 8 1 1 2 4

Matrix representation of C8×C9⋊C6 in GL6(𝔽73)

 22 0 0 0 0 0 0 22 0 0 0 0 0 0 22 0 0 0 0 0 0 22 0 0 0 0 0 0 22 0 0 0 0 0 0 22
,
 0 0 0 1 0 0 0 0 72 72 0 0 0 0 0 0 0 1 0 0 0 0 72 72 1 0 0 0 0 0 0 1 0 0 0 0
,
 1 0 0 0 0 0 72 72 0 0 0 0 0 0 0 0 1 0 0 0 0 0 72 72 0 0 72 72 0 0 0 0 0 1 0 0

G:=sub<GL(6,GF(73))| [22,0,0,0,0,0,0,22,0,0,0,0,0,0,22,0,0,0,0,0,0,22,0,0,0,0,0,0,22,0,0,0,0,0,0,22],[0,0,0,0,1,0,0,0,0,0,0,1,0,72,0,0,0,0,1,72,0,0,0,0,0,0,0,72,0,0,0,0,1,72,0,0],[1,72,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,72,1,0,0,1,72,0,0,0,0,0,72,0,0] >;

C8×C9⋊C6 in GAP, Magma, Sage, TeX

C_8\times C_9\rtimes C_6
% in TeX

G:=Group("C8xC9:C6");
// GroupNames label

G:=SmallGroup(432,120);
// by ID

G=gap.SmallGroup(432,120);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-3,-3,92,80,10085,2035,292,14118]);
// Polycyclic

G:=Group<a,b,c|a^8=b^9=c^6=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^2>;
// generators/relations

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