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## G = C62.3D4order 288 = 25·32

### 3rd non-split extension by C62 of D4 acting faithfully

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C3⋊Dic3 — C62.3D4
 Chief series C1 — C32 — C3×C6 — C3⋊Dic3 — C2×C3⋊Dic3 — C2×D6⋊S3 — C62.3D4
 Lower central C32 — C3×C6 — C3⋊Dic3 — C62.3D4
 Upper central C1 — C22

Generators and relations for C62.3D4
G = < a,b,c,d | a6=b6=d2=1, c4=b3, ab=ba, cac-1=a3b4, dad=a-1, cbc-1=a2b3, bd=db, dcd=a3b3c3 >

Subgroups: 456 in 90 conjugacy classes, 19 normal (17 characteristic)
C1, C2 [×3], C2 [×2], C3 [×2], C4 [×3], C22, C22 [×4], S3 [×2], C6 [×8], C8, C2×C4 [×2], D4 [×3], C23, C32, Dic3 [×5], C12, D6 [×4], C2×C6 [×6], C4⋊C4, C2×C8, C2×D4, C3×S3 [×2], C3×C6 [×3], C2×Dic3 [×3], C3⋊D4 [×4], C2×C12, C22×S3, C22×C6, D4⋊C4, C3×Dic3, C3⋊Dic3 [×2], S3×C6 [×4], C62, Dic3⋊C4, C2×C3⋊D4, C322C8, D6⋊S3 [×2], D6⋊S3, C6×Dic3, C2×C3⋊Dic3, S3×C2×C6, C62.C22, C2×C322C8, C2×D6⋊S3, C62.3D4
Quotients: C1, C2 [×3], C4 [×2], C22, C2×C4, D4 [×2], C22⋊C4, D8, SD16, D4⋊C4, S3≀C2, S32⋊C4, C32⋊D8, C322SD16, C62.3D4

Character table of C62.3D4

 class 1 2A 2B 2C 2D 2E 3A 3B 4A 4B 4C 4D 6A 6B 6C 6D 6E 6F 6G 6H 6I 6J 8A 8B 8C 8D 12A 12B 12C 12D size 1 1 1 1 12 12 4 4 12 12 18 18 4 4 4 4 4 4 12 12 12 12 18 18 18 18 12 12 12 12 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 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 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 -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 -1 -1 -1 -1 -1 -1 linear of order 2 ρ5 1 1 -1 -1 -1 1 1 1 i -i -1 1 -1 -1 1 -1 1 -1 1 1 -1 -1 -i i -i i -i -i i i linear of order 4 ρ6 1 1 -1 -1 1 -1 1 1 -i i -1 1 -1 -1 1 -1 1 -1 -1 -1 1 1 -i i -i i i i -i -i linear of order 4 ρ7 1 1 -1 -1 1 -1 1 1 i -i -1 1 -1 -1 1 -1 1 -1 -1 -1 1 1 i -i i -i -i -i i i linear of order 4 ρ8 1 1 -1 -1 -1 1 1 1 -i i -1 1 -1 -1 1 -1 1 -1 1 1 -1 -1 i -i i -i i i -i -i linear of order 4 ρ9 2 2 -2 -2 0 0 2 2 0 0 2 -2 -2 -2 2 -2 2 -2 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ10 2 2 2 2 0 0 2 2 0 0 -2 -2 2 2 2 2 2 2 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ11 2 -2 2 -2 0 0 2 2 0 0 0 0 -2 2 -2 -2 -2 2 0 0 0 0 -√2 -√2 √2 √2 0 0 0 0 orthogonal lifted from D8 ρ12 2 -2 2 -2 0 0 2 2 0 0 0 0 -2 2 -2 -2 -2 2 0 0 0 0 √2 √2 -√2 -√2 0 0 0 0 orthogonal lifted from D8 ρ13 2 -2 -2 2 0 0 2 2 0 0 0 0 2 -2 -2 2 -2 -2 0 0 0 0 √-2 -√-2 -√-2 √-2 0 0 0 0 complex lifted from SD16 ρ14 2 -2 -2 2 0 0 2 2 0 0 0 0 2 -2 -2 2 -2 -2 0 0 0 0 -√-2 √-2 √-2 -√-2 0 0 0 0 complex lifted from SD16 ρ15 4 4 4 4 2 2 -2 1 0 0 0 0 -2 -2 -2 1 1 1 -1 -1 -1 -1 0 0 0 0 0 0 0 0 orthogonal lifted from S3≀C2 ρ16 4 4 4 4 0 0 1 -2 2 2 0 0 1 1 1 -2 -2 -2 0 0 0 0 0 0 0 0 -1 -1 -1 -1 orthogonal lifted from S3≀C2 ρ17 4 4 4 4 0 0 1 -2 -2 -2 0 0 1 1 1 -2 -2 -2 0 0 0 0 0 0 0 0 1 1 1 1 orthogonal lifted from S3≀C2 ρ18 4 4 4 4 -2 -2 -2 1 0 0 0 0 -2 -2 -2 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 orthogonal lifted from S3≀C2 ρ19 4 4 -4 -4 2 -2 -2 1 0 0 0 0 2 2 -2 -1 1 -1 1 1 -1 -1 0 0 0 0 0 0 0 0 orthogonal lifted from S32⋊C4 ρ20 4 4 -4 -4 -2 2 -2 1 0 0 0 0 2 2 -2 -1 1 -1 -1 -1 1 1 0 0 0 0 0 0 0 0 orthogonal lifted from S32⋊C4 ρ21 4 -4 -4 4 0 0 1 -2 0 0 0 0 1 -1 -1 -2 2 2 0 0 0 0 0 0 0 0 -√3 √3 -√3 √3 symplectic lifted from C32⋊2SD16, Schur index 2 ρ22 4 -4 -4 4 0 0 1 -2 0 0 0 0 1 -1 -1 -2 2 2 0 0 0 0 0 0 0 0 √3 -√3 √3 -√3 symplectic lifted from C32⋊2SD16, Schur index 2 ρ23 4 -4 4 -4 0 0 -2 1 0 0 0 0 2 -2 2 -1 -1 1 √-3 -√-3 -√-3 √-3 0 0 0 0 0 0 0 0 complex lifted from C32⋊D8 ρ24 4 -4 -4 4 0 0 -2 1 0 0 0 0 -2 2 2 1 -1 -1 -√-3 √-3 -√-3 √-3 0 0 0 0 0 0 0 0 complex lifted from C32⋊2SD16 ρ25 4 -4 4 -4 0 0 1 -2 0 0 0 0 -1 1 -1 2 2 -2 0 0 0 0 0 0 0 0 -√-3 √-3 √-3 -√-3 complex lifted from C32⋊D8 ρ26 4 -4 -4 4 0 0 -2 1 0 0 0 0 -2 2 2 1 -1 -1 √-3 -√-3 √-3 -√-3 0 0 0 0 0 0 0 0 complex lifted from C32⋊2SD16 ρ27 4 4 -4 -4 0 0 1 -2 -2i 2i 0 0 -1 -1 1 2 -2 2 0 0 0 0 0 0 0 0 -i -i i i complex lifted from S32⋊C4 ρ28 4 4 -4 -4 0 0 1 -2 2i -2i 0 0 -1 -1 1 2 -2 2 0 0 0 0 0 0 0 0 i i -i -i complex lifted from S32⋊C4 ρ29 4 -4 4 -4 0 0 1 -2 0 0 0 0 -1 1 -1 2 2 -2 0 0 0 0 0 0 0 0 √-3 -√-3 -√-3 √-3 complex lifted from C32⋊D8 ρ30 4 -4 4 -4 0 0 -2 1 0 0 0 0 2 -2 2 -1 -1 1 -√-3 √-3 √-3 -√-3 0 0 0 0 0 0 0 0 complex lifted from C32⋊D8

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

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

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

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

Matrix representation of C62.3D4 in GL6(𝔽73)

 72 0 0 0 0 0 0 72 0 0 0 0 0 0 1 1 0 0 0 0 72 0 0 0 0 0 0 0 72 0 0 0 0 0 0 72
,
 72 0 0 0 0 0 0 72 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 72 72 0 0 0 0 1 0
,
 67 6 0 0 0 0 67 67 0 0 0 0 0 0 0 0 72 0 0 0 0 0 0 72 0 0 72 0 0 0 0 0 1 1 0 0
,
 1 0 0 0 0 0 0 72 0 0 0 0 0 0 1 0 0 0 0 0 72 72 0 0 0 0 0 0 72 0 0 0 0 0 0 72

`G:=sub<GL(6,GF(73))| [72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,72,0,0,0,0,1,0,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,1,0,0,0,0,72,0],[67,67,0,0,0,0,6,67,0,0,0,0,0,0,0,0,72,1,0,0,0,0,0,1,0,0,72,0,0,0,0,0,0,72,0,0],[1,0,0,0,0,0,0,72,0,0,0,0,0,0,1,72,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72] >;`

C62.3D4 in GAP, Magma, Sage, TeX

`C_6^2._3D_4`
`% in TeX`

`G:=Group("C6^2.3D4");`
`// GroupNames label`

`G:=SmallGroup(288,387);`
`// by ID`

`G=gap.SmallGroup(288,387);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,85,64,422,219,100,2693,2028,691,797,2372]);`
`// 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,c*b*c^-1=a^2*b^3,b*d=d*b,d*c*d=a^3*b^3*c^3>;`
`// generators/relations`

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