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## G = C62.110D4order 288 = 25·32

### 15th non-split extension by C62 of D4 acting via D4/C22=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C62 — C62.110D4
 Chief series C1 — C3 — C32 — C3×C6 — C62 — C2×C62 — C2×C32⋊7D4 — C62.110D4
 Lower central C32 — C3×C6 — C62 — C62.110D4
 Upper central C1 — C2 — C23 — C22⋊C4

Generators and relations for C62.110D4
G = < a,b,c,d | a6=b6=c4=d2=1, ab=ba, cac-1=dad=a-1b3, cbc-1=dbd=b-1, dcd=a3c-1 >

Subgroups: 732 in 156 conjugacy classes, 47 normal (19 characteristic)
C1, C2, C2 [×4], C3 [×4], C4 [×3], C22 [×3], C22 [×3], S3 [×4], C6 [×4], C6 [×12], C2×C4 [×3], D4 [×2], C23, C23, C32, Dic3 [×8], C12 [×4], D6 [×8], C2×C6 [×12], C2×C6 [×4], C22⋊C4, C22⋊C4, C2×D4, C3⋊S3, C3×C6, C3×C6 [×3], C2×Dic3 [×8], C3⋊D4 [×8], C2×C12 [×4], C22×S3 [×4], C22×C6 [×4], C23⋊C4, C3⋊Dic3 [×2], C3×C12, C2×C3⋊S3 [×2], C62 [×3], C62, C6.D4 [×4], C3×C22⋊C4 [×4], C2×C3⋊D4 [×4], C2×C3⋊Dic3, C2×C3⋊Dic3, C327D4 [×2], C6×C12, C22×C3⋊S3, C2×C62, C23.6D6 [×4], C625C4, C32×C22⋊C4, C2×C327D4, C62.110D4
Quotients: C1, C2 [×3], C4 [×2], C22, S3 [×4], C2×C4, D4 [×2], D6 [×4], C22⋊C4, C3⋊S3, C4×S3 [×4], D12 [×4], C3⋊D4 [×4], C23⋊C4, C2×C3⋊S3, D6⋊C4 [×4], C4×C3⋊S3, C12⋊S3, C327D4, C23.6D6 [×4], C6.11D12, C62.110D4

Smallest permutation representation of C62.110D4
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 54 61 16 24 58)(2 49 62 17 19 59)(3 50 63 18 20 60)(4 51 64 13 21 55)(5 52 65 14 22 56)(6 53 66 15 23 57)(7 38 30 33 72 44)(8 39 25 34 67 45)(9 40 26 35 68 46)(10 41 27 36 69 47)(11 42 28 31 70 48)(12 37 29 32 71 43)
(2 15)(3 5)(4 13)(6 17)(7 45 33 25)(8 30 34 44)(9 43 35 29)(10 28 36 48)(11 47 31 27)(12 26 32 46)(14 18)(19 57)(20 65)(21 55)(22 63)(23 59)(24 61)(37 40 71 68)(38 67 72 39)(41 42 69 70)(49 66)(50 56)(51 64)(52 60)(53 62)(54 58)
(1 37)(2 70)(3 41)(4 68)(5 39)(6 72)(7 23)(8 52)(9 21)(10 50)(11 19)(12 54)(13 40)(14 67)(15 38)(16 71)(17 42)(18 69)(20 36)(22 34)(24 32)(25 56)(26 64)(27 60)(28 62)(29 58)(30 66)(31 49)(33 53)(35 51)(43 61)(44 57)(45 65)(46 55)(47 63)(48 59)```

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

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

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

51 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3A 3B 3C 3D 4A 4B 4C 4D 4E 6A ··· 6L 6M ··· 6T 12A ··· 12P order 1 2 2 2 2 2 3 3 3 3 4 4 4 4 4 6 ··· 6 6 ··· 6 12 ··· 12 size 1 1 2 2 2 36 2 2 2 2 4 4 36 36 36 2 ··· 2 4 ··· 4 4 ··· 4

51 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 4 4 type + + + + + + + + + image C1 C2 C2 C2 C4 C4 S3 D4 D6 C4×S3 D12 C3⋊D4 C23⋊C4 C23.6D6 kernel C62.110D4 C62⋊5C4 C32×C22⋊C4 C2×C32⋊7D4 C2×C3⋊Dic3 C22×C3⋊S3 C3×C22⋊C4 C62 C22×C6 C2×C6 C2×C6 C2×C6 C32 C3 # reps 1 1 1 1 2 2 4 2 4 8 8 8 1 8

Matrix representation of C62.110D4 in GL6(𝔽13)

 0 1 0 0 0 0 12 1 0 0 0 0 0 0 2 4 2 7 0 0 9 11 7 4 0 0 0 0 11 4 0 0 0 0 9 2
,
 0 12 0 0 0 0 1 12 0 0 0 0 0 0 1 1 7 5 0 0 12 0 5 1 0 0 0 0 0 1 0 0 0 0 12 1
,
 8 5 0 0 0 0 0 5 0 0 0 0 0 0 1 1 6 9 0 0 0 12 9 10 0 0 0 0 9 2 0 0 0 0 11 4
,
 12 1 0 0 0 0 0 1 0 0 0 0 0 0 5 5 12 4 0 0 2 4 8 3 0 0 12 1 7 10 0 0 1 2 6 10

`G:=sub<GL(6,GF(13))| [0,12,0,0,0,0,1,1,0,0,0,0,0,0,2,9,0,0,0,0,4,11,0,0,0,0,2,7,11,9,0,0,7,4,4,2],[0,1,0,0,0,0,12,12,0,0,0,0,0,0,1,12,0,0,0,0,1,0,0,0,0,0,7,5,0,12,0,0,5,1,1,1],[8,0,0,0,0,0,5,5,0,0,0,0,0,0,1,0,0,0,0,0,1,12,0,0,0,0,6,9,9,11,0,0,9,10,2,4],[12,0,0,0,0,0,1,1,0,0,0,0,0,0,5,2,12,1,0,0,5,4,1,2,0,0,12,8,7,6,0,0,4,3,10,10] >;`

C62.110D4 in GAP, Magma, Sage, TeX

`C_6^2._{110}D_4`
`% in TeX`

`G:=Group("C6^2.110D4");`
`// GroupNames label`

`G:=SmallGroup(288,281);`
`// by ID`

`G=gap.SmallGroup(288,281);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,141,36,422,346,2693,9414]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^6=b^6=c^4=d^2=1,a*b=b*a,c*a*c^-1=d*a*d=a^-1*b^3,c*b*c^-1=d*b*d=b^-1,d*c*d=a^3*c^-1>;`
`// generators/relations`

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