Copied to
clipboard

## G = M4(2).15D6order 192 = 26·3

### 15th non-split extension by M4(2) of D6 acting via D6/C3=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C12 — M4(2).15D6
 Chief series C1 — C3 — C6 — C12 — C2×C12 — C2×D12 — D4⋊D6 — M4(2).15D6
 Lower central C3 — C6 — C2×C12 — M4(2).15D6
 Upper central C1 — C2 — C2×C4 — C8.C22

Generators and relations for M4(2).15D6
G = < a,b,c,d | a8=b2=c6=1, d2=a2, bab=a5, cac-1=a3, dad-1=a3b, cbc-1=dbd-1=a4b, dcd-1=a2c-1 >

Subgroups: 304 in 104 conjugacy classes, 37 normal (all characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C12, C12, D6, C2×C6, C2×C6, C2×C8, M4(2), M4(2), D8, SD16, Q16, C2×D4, C2×Q8, C4○D4, C3⋊C8, C3⋊C8, C24, D12, C2×C12, C2×C12, C3×D4, C3×D4, C3×Q8, C3×Q8, C22×S3, C4.D4, C4.10D4, C8.C4, C8○D4, C2×SD16, C8⋊C22, C8.C22, C2×C3⋊C8, C2×C3⋊C8, C4.Dic3, C4.Dic3, D4⋊S3, Q82S3, C3×M4(2), C3×SD16, C3×Q16, C2×D12, C6×Q8, C3×C4○D4, D4.3D4, C12.53D4, C12.46D4, C12.10D4, C2×Q82S3, D4.Dic3, D4⋊D6, C3×C8.C22, M4(2).15D6
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, C3⋊D4, C22×S3, C4⋊D4, S3×D4, D42S3, C2×C3⋊D4, D4.3D4, C23.14D6, M4(2).15D6

Character table of M4(2).15D6

 class 1 2A 2B 2C 2D 3 4A 4B 4C 4D 6A 6B 6C 8A 8B 8C 8D 8E 8F 8G 12A 12B 12C 12D 12E 24A 24B size 1 1 2 4 24 2 2 2 4 8 2 4 8 6 6 8 12 12 12 24 4 4 8 8 8 8 8 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 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 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 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 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 -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 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 -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 1 -1 -1 linear of order 2 ρ9 2 2 -2 0 0 2 2 -2 0 0 2 -2 0 -2 -2 0 0 0 2 0 2 -2 0 0 0 0 0 orthogonal lifted from D4 ρ10 2 2 2 -2 0 -1 2 2 -2 -2 -1 -1 1 0 0 2 0 0 0 0 -1 -1 1 1 1 -1 -1 orthogonal lifted from D6 ρ11 2 2 2 2 0 -1 2 2 2 2 -1 -1 -1 0 0 2 0 0 0 0 -1 -1 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ12 2 2 -2 0 0 2 2 -2 0 0 2 -2 0 2 2 0 0 0 -2 0 2 -2 0 0 0 0 0 orthogonal lifted from D4 ρ13 2 2 2 2 0 -1 2 2 2 -2 -1 -1 -1 0 0 -2 0 0 0 0 -1 -1 1 -1 1 1 1 orthogonal lifted from D6 ρ14 2 2 2 -2 0 -1 2 2 -2 2 -1 -1 1 0 0 -2 0 0 0 0 -1 -1 -1 1 -1 1 1 orthogonal lifted from D6 ρ15 2 2 -2 2 0 2 -2 2 -2 0 2 -2 2 0 0 0 0 0 0 0 -2 2 0 -2 0 0 0 orthogonal lifted from D4 ρ16 2 2 -2 -2 0 2 -2 2 2 0 2 -2 -2 0 0 0 0 0 0 0 -2 2 0 2 0 0 0 orthogonal lifted from D4 ρ17 2 2 -2 -2 0 -1 -2 2 2 0 -1 1 1 0 0 0 0 0 0 0 1 -1 √-3 -1 -√-3 √-3 -√-3 complex lifted from C3⋊D4 ρ18 2 2 -2 2 0 -1 -2 2 -2 0 -1 1 -1 0 0 0 0 0 0 0 1 -1 -√-3 1 √-3 √-3 -√-3 complex lifted from C3⋊D4 ρ19 2 2 -2 -2 0 -1 -2 2 2 0 -1 1 1 0 0 0 0 0 0 0 1 -1 -√-3 -1 √-3 -√-3 √-3 complex lifted from C3⋊D4 ρ20 2 2 -2 2 0 -1 -2 2 -2 0 -1 1 -1 0 0 0 0 0 0 0 1 -1 √-3 1 -√-3 -√-3 √-3 complex lifted from C3⋊D4 ρ21 2 2 2 0 0 2 -2 -2 0 0 2 2 0 0 0 0 2i -2i 0 0 -2 -2 0 0 0 0 0 complex lifted from C4○D4 ρ22 2 2 2 0 0 2 -2 -2 0 0 2 2 0 0 0 0 -2i 2i 0 0 -2 -2 0 0 0 0 0 complex lifted from C4○D4 ρ23 4 4 -4 0 0 -2 4 -4 0 0 -2 2 0 0 0 0 0 0 0 0 -2 2 0 0 0 0 0 orthogonal lifted from S3×D4 ρ24 4 4 4 0 0 -2 -4 -4 0 0 -2 -2 0 0 0 0 0 0 0 0 2 2 0 0 0 0 0 symplectic lifted from D4⋊2S3, Schur index 2 ρ25 4 -4 0 0 0 4 0 0 0 0 -4 0 0 2√-2 -2√-2 0 0 0 0 0 0 0 0 0 0 0 0 complex lifted from D4.3D4 ρ26 4 -4 0 0 0 4 0 0 0 0 -4 0 0 -2√-2 2√-2 0 0 0 0 0 0 0 0 0 0 0 0 complex lifted from D4.3D4 ρ27 8 -8 0 0 0 -4 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal faithful

Smallest permutation representation of M4(2).15D6
On 48 points
Generators in S48
```(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 6)(4 8)(10 14)(12 16)(18 22)(20 24)(26 30)(28 32)(34 38)(36 40)(41 45)(43 47)
(1 12 46 22 31 38)(2 15 47 17 32 33)(3 10 48 20 25 36)(4 13 41 23 26 39)(5 16 42 18 27 34)(6 11 43 21 28 37)(7 14 44 24 29 40)(8 9 45 19 30 35)
(1 38 3 40 5 34 7 36)(2 33 4 35 6 37 8 39)(9 28 11 30 13 32 15 26)(10 31 12 25 14 27 16 29)(17 41 19 43 21 45 23 47)(18 44 20 46 22 48 24 42)```

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

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

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

Matrix representation of M4(2).15D6 in GL8(ℤ)

 0 0 0 0 1 -1 0 -1 0 0 0 0 1 0 1 -1 0 0 0 0 0 -1 -1 1 0 0 0 0 1 -1 -1 0 0 1 1 -1 0 0 0 0 -1 1 1 0 0 0 0 0 1 -1 0 -1 0 0 0 0 1 0 1 -1 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 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 0 -1 0 0 0 0 0 0 0 0 -1
,
 0 0 0 0 0 -1 0 0 0 0 0 0 1 -1 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 1 -1 0 -1 0 0 0 0 0 0 1 -1 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 1 -1 0 0 0 0
,
 0 0 0 0 0 0 1 -1 0 0 0 0 0 0 0 -1 0 0 0 0 1 -1 0 0 0 0 0 0 0 -1 0 0 1 -1 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 -1 1 0 0 0 0 0 0 0 1 0 0 0 0

`G:=sub<GL(8,Integers())| [0,0,0,0,0,-1,1,1,0,0,0,0,1,1,-1,0,0,0,0,0,1,1,0,1,0,0,0,0,-1,0,-1,-1,1,1,0,1,0,0,0,0,-1,0,-1,-1,0,0,0,0,0,1,-1,-1,0,0,0,0,-1,-1,1,0,0,0,0,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1],[0,0,0,0,0,1,0,0,0,0,0,0,-1,-1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,-1,0,1,0,0,0,0,0,0,-1,-1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,-1,0,0,0,0],[0,0,0,0,1,0,0,0,0,0,0,0,-1,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,1,1,0,0,1,0,0,0,0,0,0,0,-1,-1,0,0,0,0,1,0,0,0,0,0,0,0,-1,-1,0,0,0,0,0,0] >;`

M4(2).15D6 in GAP, Magma, Sage, TeX

`M_4(2)._{15}D_6`
`% in TeX`

`G:=Group("M4(2).15D6");`
`// GroupNames label`

`G:=SmallGroup(192,762);`
`// by ID`

`G=gap.SmallGroup(192,762);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,253,254,219,184,1123,297,136,1684,851,438,102,6278]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^8=b^2=c^6=1,d^2=a^2,b*a*b=a^5,c*a*c^-1=a^3,d*a*d^-1=a^3*b,c*b*c^-1=d*b*d^-1=a^4*b,d*c*d^-1=a^2*c^-1>;`
`// generators/relations`

Export

׿
×
𝔽