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## G = C62.36D6order 432 = 24·33

### 19th non-split extension by C62 of D6 acting via D6/C2=S3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6 — C62.36D6
 Chief series C1 — C3 — C32 — C3×C6 — C2×He3 — C2×C32⋊C6 — C4×C32⋊C6 — C62.36D6
 Lower central C32 — C3×C6 — C62.36D6
 Upper central C1 — C4 — C2×C4

Generators and relations for C62.36D6
G = < a,b,c,d | a6=b6=1, c6=d2=b3, ab=ba, cac-1=ab2, dad-1=ab-1, bc=cb, dbd-1=b-1, dcd-1=c5 >

Subgroups: 677 in 156 conjugacy classes, 52 normal (36 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C22, S3, C6, C6, C2×C4, C2×C4, D4, Q8, C32, C32, Dic3, C12, C12, D6, C2×C6, C2×C6, C4○D4, C3×S3, C3⋊S3, C3×C6, C3×C6, Dic6, C4×S3, D12, C3⋊D4, C2×C12, C2×C12, C3×D4, C3×Q8, He3, C3×Dic3, C3⋊Dic3, C3×C12, C3×C12, S3×C6, C2×C3⋊S3, C62, C62, C4○D12, C3×C4○D4, C32⋊C6, C2×He3, C2×He3, C3×Dic6, S3×C12, C3×D12, C3×C3⋊D4, C324Q8, C4×C3⋊S3, C12⋊S3, C327D4, C6×C12, C6×C12, C32⋊C12, C4×He3, C2×C32⋊C6, C22×He3, C3×C4○D12, C12.59D6, He33Q8, C4×C32⋊C6, He34D4, He36D4, C2×C4×He3, C62.36D6
Quotients: C1, C2, C3, C22, S3, C6, C23, D6, C2×C6, C4○D4, C3×S3, C22×S3, C22×C6, S3×C6, C4○D12, C3×C4○D4, C32⋊C6, S3×C2×C6, C2×C32⋊C6, C3×C4○D12, C22×C32⋊C6, C62.36D6

Smallest permutation representation of C62.36D6
On 72 points
Generators in S72
```(2 70 60)(3 49 71)(5 61 51)(6 52 62)(8 64 54)(9 55 65)(11 67 57)(12 58 68)(13 30 45 19 36 39)(14 40 25 20 46 31)(15 21)(16 33 48 22 27 42)(17 43 28 23 37 34)(18 24)(26 32)(29 35)(38 44)(41 47)
(1 53 69 7 59 63)(2 54 70 8 60 64)(3 55 71 9 49 65)(4 56 72 10 50 66)(5 57 61 11 51 67)(6 58 62 12 52 68)(13 39 36 19 45 30)(14 40 25 20 46 31)(15 41 26 21 47 32)(16 42 27 22 48 33)(17 43 28 23 37 34)(18 44 29 24 38 35)
(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 34 7 28)(2 27 8 33)(3 32 9 26)(4 25 10 31)(5 30 11 36)(6 35 12 29)(13 61 19 67)(14 66 20 72)(15 71 21 65)(16 64 22 70)(17 69 23 63)(18 62 24 68)(37 59 43 53)(38 52 44 58)(39 57 45 51)(40 50 46 56)(41 55 47 49)(42 60 48 54)```

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

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

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

62 conjugacy classes

 class 1 2A 2B 2C 2D 3A 3B 3C 3D 3E 3F 4A 4B 4C 4D 4E 6A 6B 6C 6D 6E 6F ··· 6P 6Q 6R 6S 6T 12A 12B 12C 12D 12E 12F 12G 12H 12I ··· 12V 12W 12X 12Y 12Z order 1 2 2 2 2 3 3 3 3 3 3 4 4 4 4 4 6 6 6 6 6 6 ··· 6 6 6 6 6 12 12 12 12 12 12 12 12 12 ··· 12 12 12 12 12 size 1 1 2 18 18 2 3 3 6 6 6 1 1 2 18 18 2 2 2 3 3 6 ··· 6 18 18 18 18 2 2 2 2 3 3 3 3 6 ··· 6 18 18 18 18

62 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 6 6 6 6 type + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C3 C6 C6 C6 C6 C6 S3 D6 D6 C4○D4 C3×S3 S3×C6 S3×C6 C4○D12 C3×C4○D4 C3×C4○D12 C32⋊C6 C2×C32⋊C6 C2×C32⋊C6 C62.36D6 kernel C62.36D6 He3⋊3Q8 C4×C32⋊C6 He3⋊4D4 He3⋊6D4 C2×C4×He3 C12.59D6 C32⋊4Q8 C4×C3⋊S3 C12⋊S3 C32⋊7D4 C6×C12 C6×C12 C3×C12 C62 He3 C2×C12 C12 C2×C6 C32 C32 C3 C2×C4 C4 C22 C1 # reps 1 1 2 1 2 1 2 2 4 2 4 2 1 2 1 2 2 4 2 4 4 8 1 2 1 4

Matrix representation of C62.36D6 in GL6(𝔽13)

 1 0 0 0 0 0 0 9 0 0 0 0 0 0 3 0 0 0 0 0 0 12 0 0 0 0 0 0 4 0 0 0 0 0 0 10
,
 10 0 0 0 0 0 0 10 0 0 0 0 0 0 10 0 0 0 0 0 0 4 0 0 0 0 0 0 4 0 0 0 0 0 0 4
,
 0 0 8 0 0 0 8 0 0 0 0 0 0 8 0 0 0 0 0 0 0 0 8 0 0 0 0 0 0 8 0 0 0 8 0 0
,
 0 0 0 0 8 0 0 0 0 0 0 8 0 0 0 8 0 0 0 0 8 0 0 0 8 0 0 0 0 0 0 8 0 0 0 0

`G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,9,0,0,0,0,0,0,3,0,0,0,0,0,0,12,0,0,0,0,0,0,4,0,0,0,0,0,0,10],[10,0,0,0,0,0,0,10,0,0,0,0,0,0,10,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[0,8,0,0,0,0,0,0,8,0,0,0,8,0,0,0,0,0,0,0,0,0,0,8,0,0,0,8,0,0,0,0,0,0,8,0],[0,0,0,0,8,0,0,0,0,0,0,8,0,0,0,8,0,0,0,0,8,0,0,0,8,0,0,0,0,0,0,8,0,0,0,0] >;`

C62.36D6 in GAP, Magma, Sage, TeX

`C_6^2._{36}D_6`
`% in TeX`

`G:=Group("C6^2.36D6");`
`// GroupNames label`

`G:=SmallGroup(432,351);`
`// by ID`

`G=gap.SmallGroup(432,351);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-3,-2,-3,-3,176,590,4037,1034,14118]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^6=b^6=1,c^6=d^2=b^3,a*b=b*a,c*a*c^-1=a*b^2,d*a*d^-1=a*b^-1,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=c^5>;`
`// generators/relations`

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