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## G = C6.D36order 432 = 24·33

### 2nd non-split extension by C6 of D36 acting via D36/D18=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C36 — C6.D36
 Chief series C1 — C3 — C9 — C3×C9 — C3×C18 — C3×C36 — C3×Dic18 — C6.D36
 Lower central C3×C9 — C3×C18 — C3×C36 — C6.D36
 Upper central C1 — C2 — C4

Generators and relations for C6.D36
G = < a,b,c | a36=c2=1, b6=a18, bab-1=cac=a-1, cbc=a27b5 >

Subgroups: 704 in 76 conjugacy classes, 25 normal (all characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, C6, C6, C8, D4, Q8, C9, C9, C32, Dic3, C12, C12, D6, SD16, D9, C18, C18, C3⋊S3, C3×C6, C3⋊C8, C24, Dic6, D12, C3×Q8, C3×C9, Dic9, C36, C36, D18, C3×Dic3, C3×C12, C2×C3⋊S3, C24⋊C2, Q82S3, C9⋊S3, C3×C18, C72, Dic18, D36, C3×C3⋊C8, C3×Dic6, C12⋊S3, C3×Dic9, C3×C36, C2×C9⋊S3, C72⋊C2, C325SD16, C9×C3⋊C8, C3×Dic18, C36⋊S3, C6.D36
Quotients: C1, C2, C22, S3, D4, D6, SD16, D9, D12, C3⋊D4, D18, S32, C24⋊C2, Q82S3, D36, C3⋊D12, S3×D9, C72⋊C2, C325SD16, C3⋊D36, C6.D36

Smallest permutation representation of C6.D36
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 45 7 39 13 69 19 63 25 57 31 51)(2 44 8 38 14 68 20 62 26 56 32 50)(3 43 9 37 15 67 21 61 27 55 33 49)(4 42 10 72 16 66 22 60 28 54 34 48)(5 41 11 71 17 65 23 59 29 53 35 47)(6 40 12 70 18 64 24 58 30 52 36 46)
(1 34)(2 33)(3 32)(4 31)(5 30)(6 29)(7 28)(8 27)(9 26)(10 25)(11 24)(12 23)(13 22)(14 21)(15 20)(16 19)(17 18)(35 36)(37 53)(38 52)(39 51)(40 50)(41 49)(42 48)(43 47)(44 46)(54 72)(55 71)(56 70)(57 69)(58 68)(59 67)(60 66)(61 65)(62 64)```

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

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

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

60 conjugacy classes

 class 1 2A 2B 3A 3B 3C 4A 4B 6A 6B 6C 8A 8B 9A 9B 9C 9D 9E 9F 12A 12B 12C 12D 12E 12F 12G 18A 18B 18C 18D 18E 18F 24A 24B 24C 24D 36A ··· 36F 36G ··· 36L 72A ··· 72L order 1 2 2 3 3 3 4 4 6 6 6 8 8 9 9 9 9 9 9 12 12 12 12 12 12 12 18 18 18 18 18 18 24 24 24 24 36 ··· 36 36 ··· 36 72 ··· 72 size 1 1 108 2 2 4 2 36 2 2 4 6 6 2 2 2 4 4 4 2 2 4 4 4 36 36 2 2 2 4 4 4 6 6 6 6 2 ··· 2 4 ··· 4 6 ··· 6

60 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 type + + + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 S3 S3 D4 D6 D6 SD16 D9 C3⋊D4 D12 D18 C24⋊C2 D36 C72⋊C2 S32 Q8⋊2S3 C3⋊D12 S3×D9 C32⋊5SD16 C3⋊D36 C6.D36 kernel C6.D36 C9×C3⋊C8 C3×Dic18 C36⋊S3 Dic18 C3×C3⋊C8 C3×C18 C36 C3×C12 C3×C9 C3⋊C8 C18 C3×C6 C12 C32 C6 C3 C12 C9 C6 C4 C3 C2 C1 # reps 1 1 1 1 1 1 1 1 1 2 3 2 2 3 4 6 12 1 1 1 3 2 3 6

Matrix representation of C6.D36 in GL6(𝔽73)

 21 3 0 0 0 0 23 52 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 3 52 0 0 0 0 11 45
,
 47 18 0 0 0 0 7 26 0 0 0 0 0 0 1 72 0 0 0 0 1 0 0 0 0 0 0 0 72 0 0 0 0 0 2 1
,
 52 70 0 0 0 0 25 21 0 0 0 0 0 0 72 0 0 0 0 0 72 1 0 0 0 0 0 0 72 0 0 0 0 0 2 1

`G:=sub<GL(6,GF(73))| [21,23,0,0,0,0,3,52,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,3,11,0,0,0,0,52,45],[47,7,0,0,0,0,18,26,0,0,0,0,0,0,1,1,0,0,0,0,72,0,0,0,0,0,0,0,72,2,0,0,0,0,0,1],[52,25,0,0,0,0,70,21,0,0,0,0,0,0,72,72,0,0,0,0,0,1,0,0,0,0,0,0,72,2,0,0,0,0,0,1] >;`

C6.D36 in GAP, Magma, Sage, TeX

`C_6.D_{36}`
`% in TeX`

`G:=Group("C6.D36");`
`// GroupNames label`

`G:=SmallGroup(432,63);`
`// by ID`

`G=gap.SmallGroup(432,63);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,56,85,36,254,58,571,10085,292,14118]);`
`// Polycyclic`

`G:=Group<a,b,c|a^36=c^2=1,b^6=a^18,b*a*b^-1=c*a*c=a^-1,c*b*c=a^27*b^5>;`
`// generators/relations`

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