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## G = C36.D4order 288 = 25·32

### 7th non-split extension by C36 of D4 acting via D4/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C18 — C36.D4
 Chief series C1 — C3 — C9 — C18 — C36 — C2×C36 — C4.Dic9 — C36.D4
 Lower central C9 — C18 — C2×C18 — C36.D4
 Upper central C1 — C2 — C2×C4 — C2×D4

Generators and relations for C36.D4
G = < a,b,c | a36=1, b4=a18, c2=a9, bab-1=a-1, cac-1=a17, cbc-1=a27b3 >

Subgroups: 196 in 69 conjugacy classes, 30 normal (16 characteristic)
C1, C2, C2 [×3], C3, C4 [×2], C22, C22 [×4], C6, C6 [×3], C8 [×2], C2×C4, D4 [×2], C23 [×2], C9, C12 [×2], C2×C6, C2×C6 [×4], M4(2) [×2], C2×D4, C18, C18 [×3], C3⋊C8 [×2], C2×C12, C3×D4 [×2], C22×C6 [×2], C4.D4, C36 [×2], C2×C18, C2×C18 [×4], C4.Dic3 [×2], C6×D4, C9⋊C8 [×2], C2×C36, D4×C9 [×2], C22×C18 [×2], C12.D4, C4.Dic9 [×2], D4×C18, C36.D4
Quotients: C1, C2 [×3], C4 [×2], C22, S3, C2×C4, D4 [×2], Dic3 [×2], D6, C22⋊C4, D9, C2×Dic3, C3⋊D4 [×2], C4.D4, Dic9 [×2], D18, C6.D4, C2×Dic9, C9⋊D4 [×2], C12.D4, C18.D4, C36.D4

Smallest permutation representation of C36.D4
On 72 points
Generators in S72
```(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)
(1 68 10 59 19 50 28 41)(2 67 11 58 20 49 29 40)(3 66 12 57 21 48 30 39)(4 65 13 56 22 47 31 38)(5 64 14 55 23 46 32 37)(6 63 15 54 24 45 33 72)(7 62 16 53 25 44 34 71)(8 61 17 52 26 43 35 70)(9 60 18 51 27 42 36 69)
(1 59 10 68 19 41 28 50)(2 40 11 49 20 58 29 67)(3 57 12 66 21 39 30 48)(4 38 13 47 22 56 31 65)(5 55 14 64 23 37 32 46)(6 72 15 45 24 54 33 63)(7 53 16 62 25 71 34 44)(8 70 17 43 26 52 35 61)(9 51 18 60 27 69 36 42)```

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

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

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

51 conjugacy classes

 class 1 2A 2B 2C 2D 3 4A 4B 6A 6B 6C 6D 6E 6F 6G 8A 8B 8C 8D 9A 9B 9C 12A 12B 18A ··· 18I 18J ··· 18U 36A ··· 36F order 1 2 2 2 2 3 4 4 6 6 6 6 6 6 6 8 8 8 8 9 9 9 12 12 18 ··· 18 18 ··· 18 36 ··· 36 size 1 1 2 4 4 2 2 2 2 2 2 4 4 4 4 36 36 36 36 2 2 2 4 4 2 ··· 2 4 ··· 4 4 ··· 4

51 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 type + + + + + + - + + - + image C1 C2 C2 C4 S3 D4 D6 Dic3 D9 C3⋊D4 D18 Dic9 C9⋊D4 C4.D4 C12.D4 C36.D4 kernel C36.D4 C4.Dic9 D4×C18 C22×C18 C6×D4 C36 C2×C12 C22×C6 C2×D4 C12 C2×C4 C23 C4 C9 C3 C1 # reps 1 2 1 4 1 2 1 2 3 4 3 6 12 1 2 6

Matrix representation of C36.D4 in GL4(𝔽73) generated by

 55 34 0 0 11 18 0 0 27 18 0 4 57 18 69 0
,
 1 0 0 63 0 0 1 72 44 0 0 72 0 1 0 72
,
 1 0 63 0 0 0 72 1 0 1 72 0 44 0 72 0
`G:=sub<GL(4,GF(73))| [55,11,27,57,34,18,18,18,0,0,0,69,0,0,4,0],[1,0,44,0,0,0,0,1,0,1,0,0,63,72,72,72],[1,0,0,44,0,0,1,0,63,72,72,72,0,1,0,0] >;`

C36.D4 in GAP, Magma, Sage, TeX

`C_{36}.D_4`
`% in TeX`

`G:=Group("C36.D4");`
`// GroupNames label`

`G:=SmallGroup(288,39);`
`// by ID`

`G=gap.SmallGroup(288,39);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,28,141,219,100,675,6725,292,9414]);`
`// Polycyclic`

`G:=Group<a,b,c|a^36=1,b^4=a^18,c^2=a^9,b*a*b^-1=a^-1,c*a*c^-1=a^17,c*b*c^-1=a^27*b^3>;`
`// generators/relations`

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