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## G = D36⋊3C6order 432 = 24·33

### 3rd semidirect product of D36 and C6 acting faithfully

Series: Derived Chief Lower central Upper central

 Derived series C1 — C18 — D36⋊3C6
 Chief series C1 — C3 — C9 — C18 — C2×3- 1+2 — C2×C9⋊C6 — C4×C9⋊C6 — D36⋊3C6
 Lower central C9 — C18 — D36⋊3C6
 Upper central C1 — C2 — Q8

Generators and relations for D363C6
G = < a,b,c | a36=b2=c6=1, bab=a-1, cac-1=a29, cbc-1=a10b >

Subgroups: 502 in 126 conjugacy classes, 52 normal (19 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, C6, C6, C2×C4, D4, Q8, C9, C9, C32, Dic3, C12, C12, D6, C2×C6, C4○D4, D9, C18, C18, C3×S3, C3×C6, C4×S3, D12, C2×C12, C3×D4, C3×Q8, C3×Q8, 3- 1+2, Dic9, C36, C36, D18, C3×Dic3, C3×C12, S3×C6, Q83S3, C3×C4○D4, C9⋊C6, C2×3- 1+2, C4×D9, D36, Q8×C9, Q8×C9, S3×C12, C3×D12, Q8×C32, C9⋊C12, C4×3- 1+2, C2×C9⋊C6, Q83D9, C3×Q83S3, C4×C9⋊C6, D36⋊C3, Q8×3- 1+2, D363C6
Quotients: C1, C2, C3, C22, S3, C6, C23, D6, C2×C6, C4○D4, C3×S3, C22×S3, C22×C6, S3×C6, Q83S3, C3×C4○D4, C9⋊C6, S3×C2×C6, C2×C9⋊C6, C3×Q83S3, C22×C9⋊C6, D363C6

Smallest permutation representation of D363C6
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 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)(37 39)(40 72)(41 71)(42 70)(43 69)(44 68)(45 67)(46 66)(47 65)(48 64)(49 63)(50 62)(51 61)(52 60)(53 59)(54 58)(55 57)
(1 70)(2 39 26 51 14 63)(3 44 15 68 27 56)(4 49)(5 54 29 66 17 42)(6 59 18 47 30 71)(7 64)(8 69 32 45 20 57)(9 38 21 62 33 50)(10 43)(11 48 35 60 23 72)(12 53 24 41 36 65)(13 58)(16 37)(19 52)(22 67)(25 46)(28 61)(31 40)(34 55)```

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

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

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

50 conjugacy classes

 class 1 2A 2B 2C 2D 3A 3B 3C 4A 4B 4C 4D 4E 6A 6B 6C 6D ··· 6I 9A 9B 9C 12A 12B 12C 12D ··· 12I 12J 12K 12L 12M 18A 18B 18C 36A ··· 36I order 1 2 2 2 2 3 3 3 4 4 4 4 4 6 6 6 6 ··· 6 9 9 9 12 12 12 12 ··· 12 12 12 12 12 18 18 18 36 ··· 36 size 1 1 18 18 18 2 3 3 2 2 2 9 9 2 3 3 18 ··· 18 6 6 6 4 4 4 6 ··· 6 9 9 9 9 6 6 6 12 ··· 12

50 irreducible representations

 dim 1 1 1 1 1 1 1 1 12 2 2 2 2 2 2 4 4 6 6 type + + + + + + + + + + image C1 C2 C2 C2 C3 C6 C6 C6 D36⋊3C6 S3 D6 C4○D4 C3×S3 S3×C6 C3×C4○D4 Q8⋊3S3 C3×Q8⋊3S3 C9⋊C6 C2×C9⋊C6 kernel D36⋊3C6 C4×C9⋊C6 D36⋊C3 Q8×3- 1+2 Q8⋊3D9 C4×D9 D36 Q8×C9 C1 Q8×C32 C3×C12 3- 1+2 C3×Q8 C12 C9 C32 C3 Q8 C4 # reps 1 3 3 1 2 6 6 2 1 1 3 2 2 6 4 1 2 1 3

Matrix representation of D363C6 in GL10(𝔽37)

 1 1 1 1 0 0 0 0 0 0 36 0 36 0 0 0 0 0 0 0 35 35 36 36 0 0 0 0 0 0 2 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 36 0 0 0 0 0 0 0 0 1 0 0 0 0 0 36 0 0 0 0 0 0 0 0 0 0 36 0 0 0 0 0 0 0 0 0 0 36 0 0 0 0 0 0 0 0 0 0 36 0 0
,
 36 0 36 0 0 0 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 36 36 0 0 0 0 0 0 0 0 0 0 0 0 0 0 36 0 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 36 1 0 0 0 0 0 0 36 0 0 0 0 0 0 0 0 0 36 1 0 0 0 0
,
 0 23 0 23 0 0 0 0 0 0 23 0 23 0 0 0 0 0 0 0 0 28 0 14 0 0 0 0 0 0 28 0 14 0 0 0 0 0 0 0 0 0 0 0 0 36 0 0 0 0 0 0 0 0 36 0 0 0 0 0 0 0 0 0 0 0 0 0 0 36 0 0 0 0 0 0 0 0 36 0 0 0 0 0 0 0 36 1 0 0 0 0 0 0 0 0 0 1 0 0

`G:=sub<GL(10,GF(37))| [1,36,35,2,0,0,0,0,0,0,1,0,35,0,0,0,0,0,0,0,1,36,36,1,0,0,0,0,0,0,1,0,36,0,0,0,0,0,0,0,0,0,0,0,0,0,36,0,0,0,0,0,0,0,0,0,0,36,0,0,0,0,0,0,0,0,0,0,36,0,0,0,0,0,0,0,0,0,0,36,0,0,0,0,1,1,0,0,0,0,0,0,0,0,36,0,0,0,0,0],[36,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,36,1,1,36,0,0,0,0,0,0,0,1,0,36,0,0,0,0,0,0,0,0,0,0,0,0,0,0,36,36,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,36,36,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,36,36,0,0,0,0,0,0,0,0,0,1,0,0,0,0],[0,23,0,28,0,0,0,0,0,0,23,0,28,0,0,0,0,0,0,0,0,23,0,14,0,0,0,0,0,0,23,0,14,0,0,0,0,0,0,0,0,0,0,0,0,36,0,0,0,0,0,0,0,0,36,0,0,0,0,0,0,0,0,0,0,0,0,0,36,0,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,36,0,0,0,0,0,0,0,0,36,0,0,0] >;`

D363C6 in GAP, Magma, Sage, TeX

`D_{36}\rtimes_3C_6`
`% in TeX`

`G:=Group("D36:3C6");`
`// GroupNames label`

`G:=SmallGroup(432,371);`
`// by ID`

`G=gap.SmallGroup(432,371);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-3,-2,-3,-3,176,590,303,142,10085,1034,292,14118]);`
`// Polycyclic`

`G:=Group<a,b,c|a^36=b^2=c^6=1,b*a*b=a^-1,c*a*c^-1=a^29,c*b*c^-1=a^10*b>;`
`// generators/relations`

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