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## G = D36.C6order 432 = 24·33

### 1st non-split extension by D36 of C6 acting faithfully

Series: Derived Chief Lower central Upper central

 Derived series C1 — C36 — D36.C6
 Chief series C1 — C3 — C9 — C18 — C36 — C4×3- 1+2 — D36⋊C3 — D36.C6
 Lower central C9 — C18 — C36 — D36.C6
 Upper central C1 — C2 — C4 — Q8

Generators and relations for D36.C6
G = < a,b,c | a36=b2=1, c6=a18, bab=a-1, cac-1=a7, cbc-1=a15b >

Subgroups: 294 in 66 conjugacy classes, 26 normal (all characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, C6, C6, C8, D4, Q8, C9, C9, C32, C12, C12, D6, C2×C6, SD16, D9, C18, C18, C3×S3, C3×C6, C3⋊C8, C24, D12, C3×D4, C3×Q8, C3×Q8, 3- 1+2, C36, C36, D18, C3×C12, C3×C12, S3×C6, Q82S3, C3×SD16, C9⋊C6, C2×3- 1+2, C9⋊C8, D36, Q8×C9, Q8×C9, C3×C3⋊C8, C3×D12, Q8×C32, C4×3- 1+2, C4×3- 1+2, C2×C9⋊C6, Q82D9, C3×Q82S3, C9⋊C24, D36⋊C3, Q8×3- 1+2, D36.C6
Quotients: C1, C2, C3, C22, S3, C6, D4, D6, C2×C6, SD16, C3×S3, C3⋊D4, C3×D4, S3×C6, Q82S3, C3×SD16, C9⋊C6, C3×C3⋊D4, C2×C9⋊C6, C3×Q82S3, Dic9⋊C6, D36.C6

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

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

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

41 conjugacy classes

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

41 irreducible representations

 dim 1 1 1 1 1 1 1 1 12 2 2 2 2 2 2 2 2 2 2 4 4 6 6 6 type + + + + + + + + + + + image C1 C2 C2 C2 C3 C6 C6 C6 D36.C6 S3 D4 D6 SD16 C3×S3 C3×D4 C3⋊D4 S3×C6 C3×SD16 C3×C3⋊D4 Q8⋊2S3 C3×Q8⋊2S3 C9⋊C6 C2×C9⋊C6 Dic9⋊C6 kernel D36.C6 C9⋊C24 D36⋊C3 Q8×3- 1+2 Q8⋊2D9 C9⋊C8 D36 Q8×C9 C1 Q8×C32 C2×3- 1+2 C3×C12 3- 1+2 C3×Q8 C18 C3×C6 C12 C9 C6 C32 C3 Q8 C4 C2 # reps 1 1 1 1 2 2 2 2 1 1 1 1 2 2 2 2 2 4 4 1 2 1 1 2

Matrix representation of D36.C6 in GL10(𝔽73)

 72 72 2 1 0 0 0 0 0 0 72 72 1 2 0 0 0 0 0 0 24 23 1 1 0 0 0 0 0 0 24 24 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 72 0 0 0 0 0 0 0 0 1 0 0 0 0 0 72 0 0 0 0 0 0 0 0 0 0 72 0 0 0 0 0 0 0 0 0 0 72 0 0 0 0 0 0 0 0 0 0 72 0 0
,
 72 72 1 2 0 0 0 0 0 0 72 72 2 1 0 0 0 0 0 0 72 0 1 1 0 0 0 0 0 0 0 72 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 72 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 72 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 72 0 0 0 0
,
 41 0 32 12 0 0 0 0 0 0 41 53 20 32 0 0 0 0 0 0 3 8 20 0 0 0 0 0 0 0 10 3 32 32 0 0 0 0 0 0 0 0 0 0 43 60 0 0 0 0 0 0 0 0 13 30 0 0 0 0 0 0 0 0 0 0 43 43 0 0 0 0 0 0 0 0 30 13 0 0 0 0 0 0 0 0 0 0 60 43 0 0 0 0 0 0 0 0 30 30

`G:=sub<GL(10,GF(73))| [72,72,24,24,0,0,0,0,0,0,72,72,23,24,0,0,0,0,0,0,2,1,1,1,0,0,0,0,0,0,1,2,1,1,0,0,0,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,0,72,0,0,0,0,1,1,0,0,0,0,0,0,0,0,72,0,0,0,0,0],[72,72,72,0,0,0,0,0,0,0,72,72,0,72,0,0,0,0,0,0,1,2,1,1,0,0,0,0,0,0,2,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,0,72,0,0,0,0],[41,41,3,10,0,0,0,0,0,0,0,53,8,3,0,0,0,0,0,0,32,20,20,32,0,0,0,0,0,0,12,32,0,32,0,0,0,0,0,0,0,0,0,0,43,13,0,0,0,0,0,0,0,0,60,30,0,0,0,0,0,0,0,0,0,0,43,30,0,0,0,0,0,0,0,0,43,13,0,0,0,0,0,0,0,0,0,0,60,30,0,0,0,0,0,0,0,0,43,30] >;`

D36.C6 in GAP, Magma, Sage, TeX

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

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

`G:=SmallGroup(432,163);`
`// by ID`

`G=gap.SmallGroup(432,163);`
`# by ID`

`G:=PCGroup([7,-2,-2,-3,-2,-2,-3,-3,197,176,1011,514,80,10085,2035,292,14118]);`
`// Polycyclic`

`G:=Group<a,b,c|a^36=b^2=1,c^6=a^18,b*a*b=a^-1,c*a*c^-1=a^7,c*b*c^-1=a^15*b>;`
`// generators/relations`

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