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## G = C18.D6order 216 = 23·33

### 3rd non-split extension by C18 of D6 acting via D6/S3=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C9 — C18.D6
 Chief series C1 — C3 — C32 — C3×C9 — C3×C18 — C9×Dic3 — C18.D6
 Lower central C3×C9 — C18.D6
 Upper central C1 — C2

Generators and relations for C18.D6
G = < a,b,c | a18=c2=1, b6=a9, bab-1=cac=a-1, cbc=b5 >

Subgroups: 338 in 58 conjugacy classes, 21 normal (19 characteristic)
C1, C2, C2, C3, C3, C4, C22, S3, C6, C6, C2×C4, C9, C9, C32, Dic3, Dic3, C12, D6, D9, C18, C18, C3⋊S3, C3×C6, C4×S3, C3×C9, Dic9, C36, D18, C3×Dic3, C3×Dic3, C2×C3⋊S3, C9⋊S3, C3×C18, C4×D9, C6.D6, C3×Dic9, C9×Dic3, C2×C9⋊S3, C18.D6
Quotients: C1, C2, C4, C22, S3, C2×C4, D6, D9, C4×S3, D18, S32, C4×D9, C6.D6, S3×D9, C18.D6

Smallest permutation representation of C18.D6
On 36 points
Generators in S36
```(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 29 16 32 13 35 10 20 7 23 4 26)(2 28 17 31 14 34 11 19 8 22 5 25)(3 27 18 30 15 33 12 36 9 21 6 24)
(1 7)(2 6)(3 5)(8 18)(9 17)(10 16)(11 15)(12 14)(19 21)(22 36)(23 35)(24 34)(25 33)(26 32)(27 31)(28 30)```

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

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

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

C18.D6 is a maximal subgroup of   Dic65D9  Dic18⋊S3  Dic9.D6  C4×S3×D9  D18.3D6  Dic3.D18  D18⋊D6
C18.D6 is a maximal quotient of   C36.38D6  C36.40D6  Dic3×Dic9  C18.Dic6  C6.18D36

36 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 4A 4B 4C 4D 6A 6B 6C 9A 9B 9C 9D 9E 9F 12A 12B 12C 12D 18A 18B 18C 18D 18E 18F 36A ··· 36F order 1 2 2 2 3 3 3 4 4 4 4 6 6 6 9 9 9 9 9 9 12 12 12 12 18 18 18 18 18 18 36 ··· 36 size 1 1 27 27 2 2 4 3 3 9 9 2 2 4 2 2 2 4 4 4 6 6 18 18 2 2 2 4 4 4 6 ··· 6

36 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + + + image C1 C2 C2 C2 C4 S3 S3 D6 D6 D9 C4×S3 C4×S3 D18 C4×D9 S32 C6.D6 S3×D9 C18.D6 kernel C18.D6 C3×Dic9 C9×Dic3 C2×C9⋊S3 C9⋊S3 Dic9 C3×Dic3 C18 C3×C6 Dic3 C9 C32 C6 C3 C6 C3 C2 C1 # reps 1 1 1 1 4 1 1 1 1 3 2 2 3 6 1 1 3 3

Matrix representation of C18.D6 in GL4(𝔽37) generated by

 36 0 0 0 0 36 0 0 0 0 20 6 0 0 31 26
,
 31 6 0 0 31 0 0 0 0 0 0 1 0 0 1 0
,
 36 0 0 0 36 1 0 0 0 0 0 36 0 0 36 0
`G:=sub<GL(4,GF(37))| [36,0,0,0,0,36,0,0,0,0,20,31,0,0,6,26],[31,31,0,0,6,0,0,0,0,0,0,1,0,0,1,0],[36,36,0,0,0,1,0,0,0,0,0,36,0,0,36,0] >;`

C18.D6 in GAP, Magma, Sage, TeX

`C_{18}.D_6`
`% in TeX`

`G:=Group("C18.D6");`
`// GroupNames label`

`G:=SmallGroup(216,28);`
`// by ID`

`G=gap.SmallGroup(216,28);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-3,-3,-3,24,31,1065,453,1444,2603]);`
`// Polycyclic`

`G:=Group<a,b,c|a^18=c^2=1,b^6=a^9,b*a*b^-1=c*a*c=a^-1,c*b*c=b^5>;`
`// generators/relations`

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