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## G = C4×C9⋊C6order 216 = 23·33

### Direct product of C4 and C9⋊C6

Series: Derived Chief Lower central Upper central

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

Generators and relations for C4×C9⋊C6
G = < a,b,c | a4=b9=c6=1, ab=ba, ac=ca, cbc-1=b2 >

Smallest permutation representation of C4×C9⋊C6
On 36 points
Generators in S36
(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)
(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)
(1 11)(2 16 8 10 5 13)(3 12 6 18 9 15)(4 17)(7 14)(19 33 22 30 25 36)(20 29)(21 34 27 28 24 31)(23 35)(26 32)

G:=sub<Sym(36)| (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), (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), (1,11)(2,16,8,10,5,13)(3,12,6,18,9,15)(4,17)(7,14)(19,33,22,30,25,36)(20,29)(21,34,27,28,24,31)(23,35)(26,32)>;

G:=Group( (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), (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), (1,11)(2,16,8,10,5,13)(3,12,6,18,9,15)(4,17)(7,14)(19,33,22,30,25,36)(20,29)(21,34,27,28,24,31)(23,35)(26,32) );

G=PermutationGroup([[(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)], [(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)], [(1,11),(2,16,8,10,5,13),(3,12,6,18,9,15),(4,17),(7,14),(19,33,22,30,25,36),(20,29),(21,34,27,28,24,31),(23,35),(26,32)]])

C4×C9⋊C6 is a maximal subgroup of   C72⋊C6  D366C6  Dic182C6  D363C6
C4×C9⋊C6 is a maximal quotient of   C72⋊C6  Dic9⋊C12  D18⋊C12

40 conjugacy classes

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

40 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 6 6 6 type + + + + + + + + image C1 C2 C2 C2 C3 C4 C6 C6 C6 C12 S3 D6 C3×S3 C4×S3 S3×C6 S3×C12 C9⋊C6 C2×C9⋊C6 C4×C9⋊C6 kernel C4×C9⋊C6 C9⋊C12 C4×3- 1+2 C2×C9⋊C6 C4×D9 C9⋊C6 Dic9 C36 D18 D9 C3×C12 C3×C6 C12 C32 C6 C3 C4 C2 C1 # reps 1 1 1 1 2 4 2 2 2 8 1 1 2 2 2 4 1 1 2

Matrix representation of C4×C9⋊C6 in GL6(𝔽37)

 31 0 0 0 0 0 0 31 0 0 0 0 0 0 31 0 0 0 0 0 0 31 0 0 0 0 0 0 31 0 0 0 0 0 0 31
,
 0 0 36 1 0 0 0 0 36 0 0 0 36 36 36 36 36 35 0 0 0 0 1 36 0 0 1 0 0 1 1 0 1 0 0 1
,
 0 36 0 0 0 0 36 0 0 0 0 0 1 1 1 1 2 1 0 0 0 0 1 36 0 0 0 36 36 0 0 0 36 0 36 0

G:=sub<GL(6,GF(37))| [31,0,0,0,0,0,0,31,0,0,0,0,0,0,31,0,0,0,0,0,0,31,0,0,0,0,0,0,31,0,0,0,0,0,0,31],[0,0,36,0,0,1,0,0,36,0,0,0,36,36,36,0,1,1,1,0,36,0,0,0,0,0,36,1,0,0,0,0,35,36,1,1],[0,36,1,0,0,0,36,0,1,0,0,0,0,0,1,0,0,36,0,0,1,0,36,0,0,0,2,1,36,36,0,0,1,36,0,0] >;

C4×C9⋊C6 in GAP, Magma, Sage, TeX

C_4\times C_9\rtimes C_6
% in TeX

G:=Group("C4xC9:C6");
// GroupNames label

G:=SmallGroup(216,53);
// by ID

G=gap.SmallGroup(216,53);
# by ID

G:=PCGroup([6,-2,-2,-3,-2,-3,-3,79,3604,736,208,5189]);
// Polycyclic

G:=Group<a,b,c|a^4=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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