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## G = C62.15D4order 288 = 25·32

### 15th non-split extension by C62 of D4 acting faithfully

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C3⋊Dic3 — C62.15D4
 Chief series C1 — C32 — C3×C6 — C3⋊Dic3 — D6⋊S3 — C32⋊2SD16 — C62.15D4
 Lower central C32 — C3×C6 — C3⋊Dic3 — C62.15D4
 Upper central C1 — C2 — C22

Generators and relations for C62.15D4
G = < a,b,c,d | a6=b6=d2=1, c4=b3, ab=ba, cac-1=a3b, dad=a-1b3, cbc-1=a2b3, bd=db, dcd=c3 >

Subgroups: 464 in 99 conjugacy classes, 23 normal (15 characteristic)
C1, C2, C2 [×2], C3 [×2], C4 [×5], C22, C22, S3, C6 [×6], C8 [×2], C2×C4 [×3], D4 [×2], Q8 [×4], C32, Dic3 [×7], C12 [×3], D6, C2×C6 [×3], M4(2), SD16 [×2], Q16 [×2], C2×Q8, C4○D4, C3×S3, C3×C6, C3×C6, Dic6 [×5], C4×S3, C2×Dic3 [×4], C3⋊D4 [×2], C2×C12, C3×D4, C8.C22, C3×Dic3 [×3], C3⋊Dic3 [×2], S3×C6, C62, C2×Dic6, D42S3, C322C8 [×2], S3×Dic3, D6⋊S3, C322Q8, C322Q8 [×2], C322Q8, C6×Dic3, C3×C3⋊D4, C2×C3⋊Dic3, C322SD16 [×2], C32⋊Q16 [×2], C62.C4, D6.4D6, C2×C322Q8, C62.15D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], C23, C2×D4, C8.C22, S3≀C2, C2×S3≀C2, C62.15D4

Character table of C62.15D4

 class 1 2A 2B 2C 3A 3B 4A 4B 4C 4D 4E 6A 6B 6C 6D 6E 6F 8A 8B 12A 12B 12C 12D 12E size 1 1 2 12 4 4 12 12 12 18 18 4 4 4 4 8 24 36 36 12 12 12 12 24 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 1 1 1 1 -1 -1 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 1 linear of order 2 ρ3 1 1 -1 1 1 1 -1 1 -1 1 -1 -1 1 -1 1 -1 1 1 -1 -1 1 1 -1 -1 linear of order 2 ρ4 1 1 -1 1 1 1 1 -1 -1 1 -1 -1 1 -1 1 -1 1 -1 1 1 -1 -1 1 -1 linear of order 2 ρ5 1 1 -1 -1 1 1 -1 1 1 1 -1 -1 1 -1 1 -1 -1 -1 1 -1 1 1 -1 1 linear of order 2 ρ6 1 1 -1 -1 1 1 1 -1 1 1 -1 -1 1 -1 1 -1 -1 1 -1 1 -1 -1 1 1 linear of order 2 ρ7 1 1 1 -1 1 1 1 1 -1 1 1 1 1 1 1 1 -1 -1 -1 1 1 1 1 -1 linear of order 2 ρ8 1 1 1 -1 1 1 -1 -1 -1 1 1 1 1 1 1 1 -1 1 1 -1 -1 -1 -1 -1 linear of order 2 ρ9 2 2 -2 0 2 2 0 0 0 -2 2 -2 2 -2 2 -2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ10 2 2 2 0 2 2 0 0 0 -2 -2 2 2 2 2 2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ11 4 4 4 2 -2 1 0 0 2 0 0 -2 -2 -2 1 1 -1 0 0 0 0 0 0 -1 orthogonal lifted from S3≀C2 ρ12 4 4 -4 -2 -2 1 0 0 2 0 0 2 -2 2 1 -1 1 0 0 0 0 0 0 -1 orthogonal lifted from C2×S3≀C2 ρ13 4 4 -4 0 1 -2 -2 2 0 0 0 -1 1 -1 -2 2 0 0 0 1 -1 -1 1 0 orthogonal lifted from C2×S3≀C2 ρ14 4 4 4 -2 -2 1 0 0 -2 0 0 -2 -2 -2 1 1 1 0 0 0 0 0 0 1 orthogonal lifted from S3≀C2 ρ15 4 4 4 0 1 -2 2 2 0 0 0 1 1 1 -2 -2 0 0 0 -1 -1 -1 -1 0 orthogonal lifted from S3≀C2 ρ16 4 4 4 0 1 -2 -2 -2 0 0 0 1 1 1 -2 -2 0 0 0 1 1 1 1 0 orthogonal lifted from S3≀C2 ρ17 4 4 -4 2 -2 1 0 0 -2 0 0 2 -2 2 1 -1 -1 0 0 0 0 0 0 1 orthogonal lifted from C2×S3≀C2 ρ18 4 4 -4 0 1 -2 2 -2 0 0 0 -1 1 -1 -2 2 0 0 0 -1 1 1 -1 0 orthogonal lifted from C2×S3≀C2 ρ19 4 -4 0 0 4 4 0 0 0 0 0 0 -4 0 -4 0 0 0 0 0 0 0 0 0 symplectic lifted from C8.C22, Schur index 2 ρ20 4 -4 0 0 1 -2 0 0 0 0 0 3 -1 -3 2 0 0 0 0 -√3 √3 -√3 √3 0 symplectic faithful, Schur index 2 ρ21 4 -4 0 0 1 -2 0 0 0 0 0 3 -1 -3 2 0 0 0 0 √3 -√3 √3 -√3 0 symplectic faithful, Schur index 2 ρ22 4 -4 0 0 1 -2 0 0 0 0 0 -3 -1 3 2 0 0 0 0 √3 √3 -√3 -√3 0 symplectic faithful, Schur index 2 ρ23 4 -4 0 0 1 -2 0 0 0 0 0 -3 -1 3 2 0 0 0 0 -√3 -√3 √3 √3 0 symplectic faithful, Schur index 2 ρ24 8 -8 0 0 -4 2 0 0 0 0 0 0 4 0 -2 0 0 0 0 0 0 0 0 0 symplectic faithful, Schur index 2

Smallest permutation representation of C62.15D4
On 48 points
Generators in S48
```(1 29 11 37 24 46)(2 34)(3 48 18 39 13 31)(4 36)(5 25 15 33 20 42)(6 38)(7 44 22 35 9 27)(8 40)(10 41)(12 43)(14 45)(16 47)(17 26)(19 28)(21 30)(23 32)
(1 5)(2 16 17 6 12 21)(3 7)(4 23 14 8 19 10)(9 13)(11 15)(18 22)(20 24)(25 29)(26 38 43 30 34 47)(27 31)(28 41 36 32 45 40)(33 37)(35 39)(42 46)(44 48)
(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)
(2 4)(3 7)(6 8)(9 18)(10 21)(11 24)(12 19)(13 22)(14 17)(15 20)(16 23)(25 46)(26 41)(27 44)(28 47)(29 42)(30 45)(31 48)(32 43)(33 37)(34 40)(36 38)```

`G:=sub<Sym(48)| (1,29,11,37,24,46)(2,34)(3,48,18,39,13,31)(4,36)(5,25,15,33,20,42)(6,38)(7,44,22,35,9,27)(8,40)(10,41)(12,43)(14,45)(16,47)(17,26)(19,28)(21,30)(23,32), (1,5)(2,16,17,6,12,21)(3,7)(4,23,14,8,19,10)(9,13)(11,15)(18,22)(20,24)(25,29)(26,38,43,30,34,47)(27,31)(28,41,36,32,45,40)(33,37)(35,39)(42,46)(44,48), (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), (2,4)(3,7)(6,8)(9,18)(10,21)(11,24)(12,19)(13,22)(14,17)(15,20)(16,23)(25,46)(26,41)(27,44)(28,47)(29,42)(30,45)(31,48)(32,43)(33,37)(34,40)(36,38)>;`

`G:=Group( (1,29,11,37,24,46)(2,34)(3,48,18,39,13,31)(4,36)(5,25,15,33,20,42)(6,38)(7,44,22,35,9,27)(8,40)(10,41)(12,43)(14,45)(16,47)(17,26)(19,28)(21,30)(23,32), (1,5)(2,16,17,6,12,21)(3,7)(4,23,14,8,19,10)(9,13)(11,15)(18,22)(20,24)(25,29)(26,38,43,30,34,47)(27,31)(28,41,36,32,45,40)(33,37)(35,39)(42,46)(44,48), (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), (2,4)(3,7)(6,8)(9,18)(10,21)(11,24)(12,19)(13,22)(14,17)(15,20)(16,23)(25,46)(26,41)(27,44)(28,47)(29,42)(30,45)(31,48)(32,43)(33,37)(34,40)(36,38) );`

`G=PermutationGroup([(1,29,11,37,24,46),(2,34),(3,48,18,39,13,31),(4,36),(5,25,15,33,20,42),(6,38),(7,44,22,35,9,27),(8,40),(10,41),(12,43),(14,45),(16,47),(17,26),(19,28),(21,30),(23,32)], [(1,5),(2,16,17,6,12,21),(3,7),(4,23,14,8,19,10),(9,13),(11,15),(18,22),(20,24),(25,29),(26,38,43,30,34,47),(27,31),(28,41,36,32,45,40),(33,37),(35,39),(42,46),(44,48)], [(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)], [(2,4),(3,7),(6,8),(9,18),(10,21),(11,24),(12,19),(13,22),(14,17),(15,20),(16,23),(25,46),(26,41),(27,44),(28,47),(29,42),(30,45),(31,48),(32,43),(33,37),(34,40),(36,38)])`

Matrix representation of C62.15D4 in GL8(𝔽73)

 72 72 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 46 19 0 0 0 0 0 0 27 27 0 0 0 0 0 0 0 0 46 19 0 0 0 0 0 0 27 27
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 72 72 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 72 0 0 0 0 0 0 0 0 72 0 0 0 0 0 0 0 0 72 0 0 0 0 0 0 0 0 72
,
 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 72 72 0 0 0 0 0 0 0 0 0 0 0 0 72 71 0 0 0 0 0 0 0 1 0 0 0 0 72 0 0 0 0 0 0 0 1 1 0 0
,
 1 0 0 0 0 0 0 0 72 72 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 72 72 0 0 0 0 0 0 0 0 72 71 0 0 0 0 0 0 0 1

`G:=sub<GL(8,GF(73))| [72,1,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,46,27,0,0,0,0,0,0,19,27,0,0,0,0,0,0,0,0,46,27,0,0,0,0,0,0,19,27],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,72,1,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72],[0,0,1,72,0,0,0,0,0,0,0,72,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,72,1,0,0,0,0,0,0,0,1,0,0,0,0,72,0,0,0,0,0,0,0,71,1,0,0],[1,72,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,72,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,71,1] >;`

C62.15D4 in GAP, Magma, Sage, TeX

`C_6^2._{15}D_4`
`% in TeX`

`G:=Group("C6^2.15D4");`
`// GroupNames label`

`G:=SmallGroup(288,887);`
`// by ID`

`G=gap.SmallGroup(288,887);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,141,120,422,675,346,80,2693,2028,362,797,1203]);`
`// Polycyclic`

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

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