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## G = C62.91D6order 432 = 24·33

### 39th non-split extension by C62 of D6 acting via D6/C3=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32×C6 — C62.91D6
 Chief series C1 — C3 — C32 — C33 — C32×C6 — S3×C3×C6 — S3×C3⋊Dic3 — C62.91D6
 Lower central C33 — C32×C6 — C62.91D6
 Upper central C1 — C2 — C22

Generators and relations for C62.91D6
G = < a,b,c,d | a6=b6=c6=1, d2=b3, ab=ba, cac-1=ab3, dad-1=a-1, cbc-1=dbd-1=b-1, dcd-1=c-1 >

Subgroups: 1472 in 304 conjugacy classes, 68 normal (32 characteristic)
C1, C2, C2, C3, C3, C3, C4, C22, C22, S3, C6, C6, C6, C2×C4, D4, Q8, C32, C32, C32, Dic3, Dic3, C12, D6, D6, C2×C6, C2×C6, C2×C6, C4○D4, C3×S3, C3⋊S3, C3×C6, C3×C6, C3×C6, Dic6, C4×S3, C2×Dic3, C3⋊D4, C3⋊D4, C3×D4, C33, C3×Dic3, C3×Dic3, C3⋊Dic3, C3⋊Dic3, C3×C12, S3×C6, S3×C6, C2×C3⋊S3, C62, C62, C62, D42S3, S3×C32, C3×C3⋊S3, C32×C6, C32×C6, S3×Dic3, D6⋊S3, C322Q8, C3×C3⋊D4, C3×C3⋊D4, C324Q8, C4×C3⋊S3, C2×C3⋊Dic3, C327D4, C327D4, D4×C32, C32×Dic3, C3×C3⋊Dic3, C335C4, S3×C3×C6, C6×C3⋊S3, C3×C62, D6.4D6, C12.D6, S3×C3⋊Dic3, Dic3×C3⋊S3, C336D4, C334Q8, C32×C3⋊D4, C3×C327D4, C2×C335C4, C62.91D6
Quotients: C1, C2, C22, S3, C23, D6, C4○D4, C3⋊S3, C22×S3, S32, C2×C3⋊S3, D42S3, C2×S32, C22×C3⋊S3, S3×C3⋊S3, D6.4D6, C12.D6, C2×S3×C3⋊S3, C62.91D6

Smallest permutation representation of C62.91D6
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 22 29 19 31 34)(2 23 30 20 32 35)(3 24 28 21 33 36)(4 14 9 12 16 25)(5 15 7 10 17 26)(6 13 8 11 18 27)(37 47 53 40 44 50)(38 48 54 41 45 51)(39 43 49 42 46 52)(55 67 65 58 70 62)(56 68 66 59 71 63)(57 69 61 60 72 64)
(1 41 32 49 28 44)(2 39 33 53 29 48)(3 37 31 51 30 46)(4 72 17 55 8 63)(5 70 18 59 9 61)(6 68 16 57 7 65)(10 67 13 56 25 64)(11 71 14 60 26 62)(12 69 15 58 27 66)(19 38 23 52 36 47)(20 42 24 50 34 45)(21 40 22 54 35 43)
(1 25 19 9)(2 27 20 8)(3 26 21 7)(4 29 12 34)(5 28 10 36)(6 30 11 35)(13 23 18 32)(14 22 16 31)(15 24 17 33)(37 60 40 57)(38 59 41 56)(39 58 42 55)(43 65 46 62)(44 64 47 61)(45 63 48 66)(49 67 52 70)(50 72 53 69)(51 71 54 68)```

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

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

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

51 conjugacy classes

 class 1 2A 2B 2C 2D 3A ··· 3E 3F 3G 3H 3I 4A 4B 4C 4D 4E 6A ··· 6E 6F ··· 6V 6W 6X 6Y 6Z 6AA 12A 12B 12C 12D 12E 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 12 12 12 12 12 size 1 1 2 6 18 2 ··· 2 4 4 4 4 6 18 27 27 54 2 ··· 2 4 ··· 4 12 12 12 12 36 12 12 12 12 36

51 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + + + + + - + - image C1 C2 C2 C2 C2 C2 C2 C2 S3 S3 D6 D6 D6 D6 D6 C4○D4 S32 D4⋊2S3 C2×S32 D6.4D6 kernel C62.91D6 S3×C3⋊Dic3 Dic3×C3⋊S3 C33⋊6D4 C33⋊4Q8 C32×C3⋊D4 C3×C32⋊7D4 C2×C33⋊5C4 C3×C3⋊D4 C32⋊7D4 C3×Dic3 C3⋊Dic3 S3×C6 C2×C3⋊S3 C62 C33 C2×C6 C32 C6 C3 # reps 1 1 1 1 1 1 1 1 4 1 4 1 4 1 5 2 4 5 4 8

Matrix representation of C62.91D6 in GL8(𝔽13)

 1 5 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 9 0 0 0 0 0 0 0 0 12 1 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 12 1 0 0 0 0 0 0 12 0
,
 5 12 0 0 0 0 0 0 11 8 0 0 0 0 0 0 0 0 9 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0
,
 8 0 0 0 0 0 0 0 0 8 0 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 9 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0

`G:=sub<GL(8,GF(13))| [1,0,0,0,0,0,0,0,5,12,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,1,0],[5,11,0,0,0,0,0,0,12,8,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0],[8,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0] >;`

C62.91D6 in GAP, Magma, Sage, TeX

`C_6^2._{91}D_6`
`% in TeX`

`G:=Group("C6^2.91D6");`
`// GroupNames label`

`G:=SmallGroup(432,676);`
`// by ID`

`G=gap.SmallGroup(432,676);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,64,254,135,571,2028,14118]);`
`// Polycyclic`

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

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