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G = C62.12D4order 288 = 25·32

12nd non-split extension by C62 of D4 acting faithfully

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C3⋊Dic3 — C62.12D4
 Chief series C1 — C32 — C3×C6 — C3⋊Dic3 — D6⋊S3 — C32⋊D8 — C62.12D4
 Lower central C32 — C3×C6 — C3⋊Dic3 — C62.12D4
 Upper central C1 — C2 — C22

Generators and relations for C62.12D4
G = < a,b,c,d | a6=b6=d2=1, c4=b3, ab=ba, cac-1=a3b, dad=a-1, cbc-1=a2b3, bd=db, dcd=b3c3 >

Subgroups: 592 in 115 conjugacy classes, 23 normal (15 characteristic)
C1, C2, C2 [×4], C3 [×2], C4 [×3], C22, C22 [×5], S3 [×3], C6 [×8], C8 [×2], C2×C4 [×2], D4 [×5], Q8, C23, C32, Dic3 [×5], C12, D6 [×5], C2×C6 [×7], M4(2), D8 [×2], SD16 [×2], C2×D4, C4○D4, C3×S3 [×3], C3×C6, C3×C6, Dic6, C4×S3, C2×Dic3 [×3], C3⋊D4 [×6], C3×D4, C22×S3, C22×C6, C8⋊C22, C3×Dic3, C3⋊Dic3 [×2], S3×C6 [×5], C62, D42S3, C2×C3⋊D4, C322C8 [×2], S3×Dic3, D6⋊S3, D6⋊S3 [×2], D6⋊S3, C322Q8, C3×C3⋊D4, C2×C3⋊Dic3, S3×C2×C6, C32⋊D8 [×2], C322SD16 [×2], C62.C4, D6.4D6, C2×D6⋊S3, C62.12D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], C23, C2×D4, C8⋊C22, S3≀C2, C2×S3≀C2, C62.12D4

Character table of C62.12D4

 class 1 2A 2B 2C 2D 2E 3A 3B 4A 4B 4C 6A 6B 6C 6D 6E 6F 6G 6H 6I 6J 8A 8B 12 size 1 1 2 12 12 12 4 4 12 18 18 4 4 4 4 8 12 12 12 12 24 36 36 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 0 2 2 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 0 0 2 2 0 -2 -2 2 2 2 2 2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ11 4 4 -4 0 2 -2 -2 1 0 0 0 -1 -1 1 -2 2 1 -1 1 -1 0 0 0 0 orthogonal lifted from C2×S3≀C2 ρ12 4 4 4 0 2 2 -2 1 0 0 0 1 1 1 -2 -2 -1 -1 -1 -1 0 0 0 0 orthogonal lifted from S3≀C2 ρ13 4 -4 0 0 0 0 4 4 0 0 0 0 0 -4 -4 0 0 0 0 0 0 0 0 0 orthogonal lifted from C8⋊C22 ρ14 4 4 -4 0 -2 2 -2 1 0 0 0 -1 -1 1 -2 2 -1 1 -1 1 0 0 0 0 orthogonal lifted from C2×S3≀C2 ρ15 4 4 -4 -2 0 0 1 -2 2 0 0 2 2 -2 1 -1 0 0 0 0 1 0 0 -1 orthogonal lifted from C2×S3≀C2 ρ16 4 4 -4 2 0 0 1 -2 -2 0 0 2 2 -2 1 -1 0 0 0 0 -1 0 0 1 orthogonal lifted from C2×S3≀C2 ρ17 4 4 4 2 0 0 1 -2 2 0 0 -2 -2 -2 1 1 0 0 0 0 -1 0 0 -1 orthogonal lifted from S3≀C2 ρ18 4 4 4 -2 0 0 1 -2 -2 0 0 -2 -2 -2 1 1 0 0 0 0 1 0 0 1 orthogonal lifted from S3≀C2 ρ19 4 4 4 0 -2 -2 -2 1 0 0 0 1 1 1 -2 -2 1 1 1 1 0 0 0 0 orthogonal lifted from S3≀C2 ρ20 4 -4 0 0 0 0 -2 1 0 0 0 3 -3 -1 2 0 √-3 -√-3 -√-3 √-3 0 0 0 0 complex faithful ρ21 4 -4 0 0 0 0 -2 1 0 0 0 3 -3 -1 2 0 -√-3 √-3 √-3 -√-3 0 0 0 0 complex faithful ρ22 4 -4 0 0 0 0 -2 1 0 0 0 -3 3 -1 2 0 -√-3 -√-3 √-3 √-3 0 0 0 0 complex faithful ρ23 4 -4 0 0 0 0 -2 1 0 0 0 -3 3 -1 2 0 √-3 √-3 -√-3 -√-3 0 0 0 0 complex faithful ρ24 8 -8 0 0 0 0 2 -4 0 0 0 0 0 4 -2 0 0 0 0 0 0 0 0 0 symplectic faithful, Schur index 2

Permutation representations of C62.12D4
On 24 points - transitive group 24T601
Generators in S24
```(1 5)(2 20 11)(3 7)(4 13 22)(6 24 15)(8 9 18)(10 14)(12 16)(17 21)(19 23)
(1 23 10 5 19 14)(2 6)(3 16 21 7 12 17)(4 8)(9 13)(11 15)(18 22)(20 24)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)(17 18 19 20 21 22 23 24)
(2 8)(3 7)(4 6)(9 11)(12 16)(13 15)(17 21)(18 20)(22 24)```

`G:=sub<Sym(24)| (1,5)(2,20,11)(3,7)(4,13,22)(6,24,15)(8,9,18)(10,14)(12,16)(17,21)(19,23), (1,23,10,5,19,14)(2,6)(3,16,21,7,12,17)(4,8)(9,13)(11,15)(18,22)(20,24), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24), (2,8)(3,7)(4,6)(9,11)(12,16)(13,15)(17,21)(18,20)(22,24)>;`

`G:=Group( (1,5)(2,20,11)(3,7)(4,13,22)(6,24,15)(8,9,18)(10,14)(12,16)(17,21)(19,23), (1,23,10,5,19,14)(2,6)(3,16,21,7,12,17)(4,8)(9,13)(11,15)(18,22)(20,24), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24), (2,8)(3,7)(4,6)(9,11)(12,16)(13,15)(17,21)(18,20)(22,24) );`

`G=PermutationGroup([(1,5),(2,20,11),(3,7),(4,13,22),(6,24,15),(8,9,18),(10,14),(12,16),(17,21),(19,23)], [(1,23,10,5,19,14),(2,6),(3,16,21,7,12,17),(4,8),(9,13),(11,15),(18,22),(20,24)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16),(17,18,19,20,21,22,23,24)], [(2,8),(3,7),(4,6),(9,11),(12,16),(13,15),(17,21),(18,20),(22,24)])`

`G:=TransitiveGroup(24,601);`

On 24 points - transitive group 24T603
Generators in S24
```(1 14 23 5 10 19)(3 21 12 7 17 16)
(1 5)(2 20 11 6 24 15)(3 7)(4 9 18 8 13 22)(10 14)(12 16)(17 21)(19 23)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)(17 18 19 20 21 22 23 24)
(2 8)(3 7)(4 6)(9 24)(10 23)(11 22)(12 21)(13 20)(14 19)(15 18)(16 17)```

`G:=sub<Sym(24)| (1,14,23,5,10,19)(3,21,12,7,17,16), (1,5)(2,20,11,6,24,15)(3,7)(4,9,18,8,13,22)(10,14)(12,16)(17,21)(19,23), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24), (2,8)(3,7)(4,6)(9,24)(10,23)(11,22)(12,21)(13,20)(14,19)(15,18)(16,17)>;`

`G:=Group( (1,14,23,5,10,19)(3,21,12,7,17,16), (1,5)(2,20,11,6,24,15)(3,7)(4,9,18,8,13,22)(10,14)(12,16)(17,21)(19,23), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24), (2,8)(3,7)(4,6)(9,24)(10,23)(11,22)(12,21)(13,20)(14,19)(15,18)(16,17) );`

`G=PermutationGroup([(1,14,23,5,10,19),(3,21,12,7,17,16)], [(1,5),(2,20,11,6,24,15),(3,7),(4,9,18,8,13,22),(10,14),(12,16),(17,21),(19,23)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16),(17,18,19,20,21,22,23,24)], [(2,8),(3,7),(4,6),(9,24),(10,23),(11,22),(12,21),(13,20),(14,19),(15,18),(16,17)])`

`G:=TransitiveGroup(24,603);`

Matrix representation of C62.12D4 in GL4(𝔽7) generated by

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

C62.12D4 in GAP, Magma, Sage, TeX

`C_6^2._{12}D_4`
`% in TeX`

`G:=Group("C6^2.12D4");`
`// GroupNames label`

`G:=SmallGroup(288,884);`
`// by ID`

`G=gap.SmallGroup(288,884);`
`# by ID`

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