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## G = C62.90D6order 432 = 24·33

### 38th non-split extension by C62 of D6 acting via D6/C3=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32×C6 — C62.90D6
 Chief series C1 — C3 — C32 — C33 — C32×C6 — S3×C3×C6 — S3×C3⋊Dic3 — C62.90D6
 Lower central C33 — C32×C6 — C62.90D6
 Upper central C1 — C2 — C22

Generators and relations for C62.90D6
G = < a,b,c,d | a6=b6=c6=1, d2=b3, ab=ba, cac-1=ab3, dad-1=a-1b3, cbc-1=dbd-1=b-1, dcd-1=c-1 >

Subgroups: 1752 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, D12, C2×Dic3, C3⋊D4, C3⋊D4, C2×C12, C3×D4, C33, C3×Dic3, C3×Dic3, C3⋊Dic3, C3⋊Dic3, C3×C12, S3×C6, C2×C3⋊S3, C62, C62, C62, C4○D12, D42S3, S3×C32, C33⋊C2, C32×C6, C32×C6, S3×Dic3, C6.D6, C3⋊D12, C322Q8, C6×Dic3, C3×C3⋊D4, C324Q8, C4×C3⋊S3, C2×C3⋊Dic3, C2×C3⋊Dic3, C327D4, D4×C32, C32×Dic3, C3×C3⋊Dic3, C335C4, S3×C3×C6, C2×C33⋊C2, C3×C62, D6.3D6, C12.D6, S3×C3⋊Dic3, C338(C2×C4), C337D4, C334Q8, C32×C3⋊D4, C6×C3⋊Dic3, C3315D4, C62.90D6
Quotients: C1, C2, C22, S3, C23, D6, C4○D4, C3⋊S3, C22×S3, S32, C2×C3⋊S3, C4○D12, D42S3, C2×S32, C22×C3⋊S3, S3×C3⋊S3, D6.3D6, C12.D6, C2×S3×C3⋊S3, C62.90D6

Smallest permutation representation of C62.90D6
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 20 31 36 27 24)(2 21 32 34 25 22)(3 19 33 35 26 23)(4 29 16 12 8 15)(5 30 17 10 9 13)(6 28 18 11 7 14)(37 53 47 40 50 44)(38 54 48 41 51 45)(39 49 43 42 52 46)(55 64 69 58 61 72)(56 65 70 59 62 67)(57 66 71 60 63 68)
(1 58 31 64 27 72)(2 56 32 62 25 70)(3 60 33 66 26 68)(4 46 8 49 16 42)(5 44 9 53 17 40)(6 48 7 51 18 38)(10 47 30 50 13 37)(11 45 28 54 14 41)(12 43 29 52 15 39)(19 71 35 57 23 63)(20 69 36 55 24 61)(21 67 34 59 22 65)
(1 45 36 48)(2 47 34 44)(3 43 35 46)(4 60 12 57)(5 56 10 59)(6 58 11 55)(7 72 28 69)(8 68 29 71)(9 70 30 67)(13 65 17 62)(14 61 18 64)(15 63 16 66)(19 49 26 52)(20 51 27 54)(21 53 25 50)(22 40 32 37)(23 42 33 39)(24 38 31 41)```

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

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

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

54 conjugacy classes

 class 1 2A 2B 2C 2D 3A ··· 3E 3F 3G 3H 3I 4A 4B 4C 4D 4E 6A ··· 6G 6H ··· 6W 6X 6Y 6Z 6AA 12A 12B 12C 12D 12E 12F 12G 12H 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 12 12 12 12 12 12 12 12 size 1 1 2 6 54 2 ··· 2 4 4 4 4 6 9 9 18 54 2 ··· 2 4 ··· 4 12 12 12 12 12 12 12 12 18 18 18 18

54 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 C4○D4 C4○D12 S32 D4⋊2S3 C2×S32 D6.3D6 kernel C62.90D6 S3×C3⋊Dic3 C33⋊8(C2×C4) C33⋊7D4 C33⋊4Q8 C32×C3⋊D4 C6×C3⋊Dic3 C33⋊15D4 C3×C3⋊D4 C2×C3⋊Dic3 C3×Dic3 C3⋊Dic3 S3×C6 C62 C33 C32 C2×C6 C32 C6 C3 # reps 1 1 1 1 1 1 1 1 4 1 4 2 4 5 2 4 4 4 4 8

Matrix representation of C62.90D6 in GL8(𝔽13)

 1 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 2 3 0 0 0 0 0 0 12 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 12 12
,
 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 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
,
 0 1 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 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 12 12
,
 0 5 0 0 0 0 0 0 5 0 0 0 0 0 0 0 0 0 4 4 0 0 0 0 0 0 6 9 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0

`G:=sub<GL(8,GF(13))| [1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,2,12,0,0,0,0,0,0,3,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,1,12],[12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,1,12],[0,5,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,0,4,6,0,0,0,0,0,0,4,9,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0] >;`

C62.90D6 in GAP, Magma, Sage, TeX

`C_6^2._{90}D_6`
`% in TeX`

`G:=Group("C6^2.90D6");`
`// GroupNames label`

`G:=SmallGroup(432,675);`
`// by ID`

`G=gap.SmallGroup(432,675);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,64,254,571,2028,14118]);`
`// 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*b^3,c*b*c^-1=d*b*d^-1=b^-1,d*c*d^-1=c^-1>;`
`// generators/relations`

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