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G = D36⋊C3order 216 = 23·33

The semidirect product of D36 and C3 acting faithfully

Series: Derived Chief Lower central Upper central

 Derived series C1 — C18 — D36⋊C3
 Chief series C1 — C3 — C9 — C18 — C2×3- 1+2 — C2×C9⋊C6 — D36⋊C3
 Lower central C9 — C18 — D36⋊C3
 Upper central C1 — C2 — C4

Generators and relations for D36⋊C3
G = < a,b,c | a36=b2=c3=1, bab=a-1, cac-1=a13, cbc-1=a12b >

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

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

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

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

D36⋊C3 is a maximal subgroup of   C722C6  D72⋊C3  D36⋊C6  D36.C6  D366C6  D4×C9⋊C6  D363C6
D36⋊C3 is a maximal quotient of   C72.C6  C722C6  D72⋊C3  C36⋊C12  D18⋊C12

31 conjugacy classes

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

31 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 6 6 6 type + + + + + + + + + + image C1 C2 C2 C3 C6 C6 S3 D4 D6 C3×S3 C3×D4 D12 S3×C6 C3×D12 C9⋊C6 C2×C9⋊C6 D36⋊C3 kernel D36⋊C3 C4×3- 1+2 C2×C9⋊C6 D36 C36 D18 C3×C12 3- 1+2 C3×C6 C12 C9 C32 C6 C3 C4 C2 C1 # reps 1 1 2 2 2 4 1 1 1 2 2 2 2 4 1 1 2

Matrix representation of D36⋊C3 in GL8(𝔽37)

 27 5 0 0 0 0 0 0 32 32 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 36 36 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0
,
 36 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0
,
 26 0 0 0 0 0 0 0 0 26 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 36 36 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 36 36

`G:=sub<GL(8,GF(37))| [27,32,0,0,0,0,0,0,5,32,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,36,0,0,0,0,0,0,1,36,0,0,0,0],[36,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0],[26,0,0,0,0,0,0,0,0,26,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,36,1,0,0,0,0,0,0,36,0,0,0,0,0,0,0,0,0,0,36,0,0,0,0,0,0,1,36] >;`

D36⋊C3 in GAP, Magma, Sage, TeX

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

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

`G:=SmallGroup(216,54);`
`// by ID`

`G=gap.SmallGroup(216,54);`
`# by ID`

`G:=PCGroup([6,-2,-2,-3,-2,-3,-3,169,79,3604,736,208,5189]);`
`// Polycyclic`

`G:=Group<a,b,c|a^36=b^2=c^3=1,b*a*b=a^-1,c*a*c^-1=a^13,c*b*c^-1=a^12*b>;`
`// generators/relations`

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