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## G = C62.13D4order 288 = 25·32

### 13rd non-split extension by C62 of D4 acting faithfully

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C3⋊Dic3 — C62.13D4
 Chief series C1 — C32 — C3×C6 — C3⋊Dic3 — D6⋊S3 — C32⋊D8 — C62.13D4
 Lower central C32 — C3×C6 — C3⋊Dic3 — C62.13D4
 Upper central C1 — C2 — C22

Generators and relations for C62.13D4
G = < a,b,c,d | a6=b6=d2=1, c4=b3, ab=ba, cac-1=a3b4, dad=a-1b3, cbc-1=a2b3, bd=db, dcd=b3c3 >

Subgroups: 496 in 102 conjugacy classes, 23 normal (15 characteristic)
C1, C2, C2 [×3], C3 [×2], C4 [×4], C22, C22 [×2], S3 [×2], C6 [×6], C8 [×2], C2×C4 [×3], D4 [×4], Q8 [×2], C32, Dic3 [×6], C12 [×2], D6 [×2], C2×C6 [×4], C2×C8, D8, SD16 [×2], Q16, C4○D4 [×2], C3×S3 [×2], C3×C6, C3×C6, Dic6 [×2], C4×S3 [×2], C2×Dic3 [×4], C3⋊D4 [×4], C3×D4 [×2], C4○D8, C3×Dic3 [×2], C3⋊Dic3 [×2], S3×C6 [×2], C62, D42S3 [×2], C322C8 [×2], S3×Dic3 [×2], D6⋊S3 [×2], C322Q8 [×2], C3×C3⋊D4 [×2], C2×C3⋊Dic3, C32⋊D8, C322SD16 [×2], C32⋊Q16, C2×C322C8, D6.4D6 [×2], C62.13D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], C23, C2×D4, C4○D8, S3≀C2, C2×S3≀C2, C62.13D4

Character table of C62.13D4

 class 1 2A 2B 2C 2D 3A 3B 4A 4B 4C 4D 4E 6A 6B 6C 6D 6E 6F 8A 8B 8C 8D 12A 12B size 1 1 2 12 12 4 4 9 9 12 12 18 4 4 8 8 24 24 18 18 18 18 24 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 0 2 2 2 2 0 0 -2 2 2 -2 -2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ10 2 2 2 0 0 2 2 -2 -2 0 0 -2 2 2 2 2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ11 2 -2 0 0 0 2 2 2i -2i 0 0 0 -2 -2 0 0 0 0 -√-2 -√2 √-2 √2 0 0 complex lifted from C4○D8 ρ12 2 -2 0 0 0 2 2 -2i 2i 0 0 0 -2 -2 0 0 0 0 -√-2 √2 √-2 -√2 0 0 complex lifted from C4○D8 ρ13 2 -2 0 0 0 2 2 2i -2i 0 0 0 -2 -2 0 0 0 0 √-2 √2 -√-2 -√2 0 0 complex lifted from C4○D8 ρ14 2 -2 0 0 0 2 2 -2i 2i 0 0 0 -2 -2 0 0 0 0 √-2 -√2 -√-2 √2 0 0 complex lifted from C4○D8 ρ15 4 4 4 0 -2 1 -2 0 0 -2 0 0 -2 1 -2 1 1 0 0 0 0 0 0 1 orthogonal lifted from S3≀C2 ρ16 4 4 -4 0 -2 1 -2 0 0 2 0 0 -2 1 2 -1 1 0 0 0 0 0 0 -1 orthogonal lifted from C2×S3≀C2 ρ17 4 4 -4 -2 0 -2 1 0 0 0 2 0 1 -2 -1 2 0 1 0 0 0 0 -1 0 orthogonal lifted from C2×S3≀C2 ρ18 4 4 4 2 0 -2 1 0 0 0 2 0 1 -2 1 -2 0 -1 0 0 0 0 -1 0 orthogonal lifted from S3≀C2 ρ19 4 4 -4 0 2 1 -2 0 0 -2 0 0 -2 1 2 -1 -1 0 0 0 0 0 0 1 orthogonal lifted from C2×S3≀C2 ρ20 4 4 4 0 2 1 -2 0 0 2 0 0 -2 1 -2 1 -1 0 0 0 0 0 0 -1 orthogonal lifted from S3≀C2 ρ21 4 4 -4 2 0 -2 1 0 0 0 -2 0 1 -2 -1 2 0 -1 0 0 0 0 1 0 orthogonal lifted from C2×S3≀C2 ρ22 4 4 4 -2 0 -2 1 0 0 0 -2 0 1 -2 1 -2 0 1 0 0 0 0 1 0 orthogonal lifted from S3≀C2 ρ23 8 -8 0 0 0 2 -4 0 0 0 0 0 4 -2 0 0 0 0 0 0 0 0 0 0 symplectic faithful, Schur index 2 ρ24 8 -8 0 0 0 -4 2 0 0 0 0 0 -2 4 0 0 0 0 0 0 0 0 0 0 symplectic faithful, Schur index 2

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

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

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

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

Matrix representation of C62.13D4 in GL6(𝔽73)

 27 54 0 0 0 0 46 46 0 0 0 0 0 0 1 72 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 72 1
,
 72 0 0 0 0 0 0 72 0 0 0 0 0 0 0 72 0 0 0 0 1 72 0 0 0 0 0 0 0 72 0 0 0 0 1 72
,
 41 41 0 0 0 0 16 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 72 0 0 0 0 0 0 72 0 0
,
 41 41 0 0 0 0 16 32 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 1 0 0

`G:=sub<GL(6,GF(73))| [27,46,0,0,0,0,54,46,0,0,0,0,0,0,1,1,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,1,1],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,1,0,0,0,0,72,72,0,0,0,0,0,0,0,1,0,0,0,0,72,72],[41,16,0,0,0,0,41,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,1,0,0,0,0,1,0,0,0],[41,16,0,0,0,0,41,32,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0] >;`

C62.13D4 in GAP, Magma, Sage, TeX

`C_6^2._{13}D_4`
`% in TeX`

`G:=Group("C6^2.13D4");`
`// GroupNames label`

`G:=SmallGroup(288,885);`
`// by ID`

`G=gap.SmallGroup(288,885);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,141,422,219,675,346,80,2693,2028,362,797,1203]);`
`// 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^4,d*a*d=a^-1*b^3,c*b*c^-1=a^2*b^3,b*d=d*b,d*c*d=b^3*c^3>;`
`// generators/relations`

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