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G = C3⋊D36order 216 = 23·33

The semidirect product of C3 and D36 acting via D36/D18=C2

Aliases: C32D36, Dic3⋊D9, D181S3, C18.4D6, C6.4D18, C32.2D12, C6.4S32, (C3×C9)⋊1D4, (C6×D9)⋊1C2, C2.5(S3×D9), C91(C3⋊D4), (C3×C6).25D6, (C9×Dic3)⋊1C2, (C3×C18).4C22, (C3×Dic3).2S3, C3.1(C3⋊D12), (C2×C9⋊S3)⋊1C2, SmallGroup(216,29)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C18 — C3⋊D36
 Chief series C1 — C3 — C32 — C3×C9 — C3×C18 — C9×Dic3 — C3⋊D36
 Lower central C3×C9 — C3×C18 — C3⋊D36
 Upper central C1 — C2

Generators and relations for C3⋊D36
G = < a,b,c | a3=b36=c2=1, bab-1=cac=a-1, cbc=b-1 >

Subgroups: 390 in 58 conjugacy classes, 19 normal (all characteristic)
C1, C2, C2, C3, C3, C4, C22, S3, C6, C6, D4, C9, C9, C32, Dic3, C12, D6, C2×C6, D9, C18, C18, C3×S3, C3⋊S3, C3×C6, D12, C3⋊D4, C3×C9, C36, D18, D18, C3×Dic3, S3×C6, C2×C3⋊S3, C3×D9, C9⋊S3, C3×C18, D36, C3⋊D12, C9×Dic3, C6×D9, C2×C9⋊S3, C3⋊D36
Quotients: C1, C2, C22, S3, D4, D6, D9, D12, C3⋊D4, D18, S32, D36, C3⋊D12, S3×D9, C3⋊D36

Smallest permutation representation of C3⋊D36
On 36 points
Generators in S36
```(1 13 25)(2 26 14)(3 15 27)(4 28 16)(5 17 29)(6 30 18)(7 19 31)(8 32 20)(9 21 33)(10 34 22)(11 23 35)(12 36 24)
(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 9)(2 8)(3 7)(4 6)(10 36)(11 35)(12 34)(13 33)(14 32)(15 31)(16 30)(17 29)(18 28)(19 27)(20 26)(21 25)(22 24)```

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

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

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

C3⋊D36 is a maximal subgroup of   D18.D6  Dic65D9  D6.D18  S3×D36  D18.3D6  D9×C3⋊D4  D18⋊D6
C3⋊D36 is a maximal quotient of   D36.S3  C6.D36  C3⋊D72  C3⋊Dic36  Dic3⋊Dic9  D18⋊Dic3  C6.18D36

33 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 4 6A 6B 6C 6D 6E 9A 9B 9C 9D 9E 9F 12A 12B 18A 18B 18C 18D 18E 18F 36A ··· 36F order 1 2 2 2 3 3 3 4 6 6 6 6 6 9 9 9 9 9 9 12 12 18 18 18 18 18 18 36 ··· 36 size 1 1 18 54 2 2 4 6 2 2 4 18 18 2 2 2 4 4 4 6 6 2 2 2 4 4 4 6 ··· 6

33 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 S3 S3 D4 D6 D6 D9 C3⋊D4 D12 D18 D36 S32 C3⋊D12 S3×D9 C3⋊D36 kernel C3⋊D36 C9×Dic3 C6×D9 C2×C9⋊S3 D18 C3×Dic3 C3×C9 C18 C3×C6 Dic3 C9 C32 C6 C3 C6 C3 C2 C1 # reps 1 1 1 1 1 1 1 1 1 3 2 2 3 6 1 1 3 3

Matrix representation of C3⋊D36 in GL4(𝔽37) generated by

 0 36 0 0 1 36 0 0 0 0 1 0 0 0 0 1
,
 14 30 0 0 7 23 0 0 0 0 31 20 0 0 17 11
,
 0 1 0 0 1 0 0 0 0 0 6 17 0 0 11 31
`G:=sub<GL(4,GF(37))| [0,1,0,0,36,36,0,0,0,0,1,0,0,0,0,1],[14,7,0,0,30,23,0,0,0,0,31,17,0,0,20,11],[0,1,0,0,1,0,0,0,0,0,6,11,0,0,17,31] >;`

C3⋊D36 in GAP, Magma, Sage, TeX

`C_3\rtimes D_{36}`
`% in TeX`

`G:=Group("C3:D36");`
`// GroupNames label`

`G:=SmallGroup(216,29);`
`// by ID`

`G=gap.SmallGroup(216,29);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-3,-3,-3,24,73,1065,453,1444,2603]);`
`// Polycyclic`

`G:=Group<a,b,c|a^3=b^36=c^2=1,b*a*b^-1=c*a*c=a^-1,c*b*c=b^-1>;`
`// generators/relations`

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