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G = M4(2).10D4order 128 = 27

10th non-split extension by M4(2) of D4 acting via D4/C2=C22

p-group, metabelian, nilpotent (class 3), monomial

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C22×C4 — M4(2).10D4
 Chief series C1 — C2 — C22 — C2×C4 — C22×C4 — C2×M4(2) — C2×C4.D4 — M4(2).10D4
 Lower central C1 — C2 — C22×C4 — M4(2).10D4
 Upper central C1 — C22 — C22×C4 — M4(2).10D4
 Jennings C1 — C2 — C2 — C22×C4 — M4(2).10D4

Generators and relations for M4(2).10D4
G = < a,b,c,d | a8=b2=c4=d2=1, bab=a5, cac-1=a5b, dad=a-1, bc=cb, dbd=a4b, dcd=a2c-1 >

Subgroups: 352 in 141 conjugacy classes, 42 normal (38 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, D4, Q8, C23, C23, C4⋊C4, C2×C8, C2×C8, M4(2), M4(2), D8, SD16, C22×C4, C22×C4, C2×D4, C2×D4, C2×D4, C2×Q8, C4○D4, C24, C22⋊C8, C4.D4, D4⋊C4, Q8⋊C4, C8.C4, C2×C4⋊C4, C22×C8, C2×M4(2), C2×D8, C2×SD16, C8⋊C22, C22×D4, C2×C4○D4, C22.C42, (C22×C8)⋊C2, C2×C4.D4, C2×D4⋊C4, C23.36D4, C2×C8.C4, C2×C8⋊C22, M4(2).10D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C22≀C2, C4⋊D4, C22.D4, C4.4D4, C23.10D4, D4.3D4, D4.4D4, M4(2).10D4

Character table of M4(2).10D4

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 4A 4B 4C 4D 4E 4F 4G 8A 8B 8C 8D 8E 8F 8G 8H 8I 8J size 1 1 1 1 2 2 8 8 8 2 2 2 2 8 8 8 4 4 4 4 8 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 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 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 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 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 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 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 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 linear of order 2 ρ9 2 2 2 2 -2 -2 -2 0 0 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 -2 2 0 2 -2 2 2 -2 -2 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ11 2 -2 2 -2 2 -2 0 0 0 2 -2 -2 2 0 0 0 0 0 0 0 -2 0 0 2 0 0 orthogonal lifted from D4 ρ12 2 2 2 2 -2 -2 0 0 0 -2 2 -2 2 0 0 0 2 -2 -2 2 0 0 0 0 0 0 orthogonal lifted from D4 ρ13 2 -2 2 -2 2 -2 0 0 0 2 -2 -2 2 0 0 0 0 0 0 0 2 0 0 -2 0 0 orthogonal lifted from D4 ρ14 2 -2 2 -2 -2 2 0 -2 2 2 2 -2 -2 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ15 2 2 2 2 -2 -2 0 0 0 -2 2 -2 2 0 0 0 -2 2 2 -2 0 0 0 0 0 0 orthogonal lifted from D4 ρ16 2 2 2 2 -2 -2 2 0 0 2 -2 2 -2 -2 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ17 2 -2 2 -2 -2 2 0 0 0 -2 -2 2 2 0 0 0 0 0 0 0 0 2i 0 0 0 -2i complex lifted from C4○D4 ρ18 2 -2 2 -2 2 -2 0 0 0 -2 2 2 -2 0 -2i 2i 0 0 0 0 0 0 0 0 0 0 complex lifted from C4○D4 ρ19 2 -2 2 -2 -2 2 0 0 0 -2 -2 2 2 0 0 0 0 0 0 0 0 -2i 0 0 0 2i complex lifted from C4○D4 ρ20 2 2 2 2 2 2 0 0 0 -2 -2 -2 -2 0 0 0 0 0 0 0 0 0 2i 0 -2i 0 complex lifted from C4○D4 ρ21 2 -2 2 -2 2 -2 0 0 0 -2 2 2 -2 0 2i -2i 0 0 0 0 0 0 0 0 0 0 complex lifted from C4○D4 ρ22 2 2 2 2 2 2 0 0 0 -2 -2 -2 -2 0 0 0 0 0 0 0 0 0 -2i 0 2i 0 complex lifted from C4○D4 ρ23 4 4 -4 -4 0 0 0 0 0 0 0 0 0 0 0 0 0 -2√2 2√2 0 0 0 0 0 0 0 orthogonal lifted from D4.4D4 ρ24 4 4 -4 -4 0 0 0 0 0 0 0 0 0 0 0 0 0 2√2 -2√2 0 0 0 0 0 0 0 orthogonal lifted from D4.4D4 ρ25 4 -4 -4 4 0 0 0 0 0 0 0 0 0 0 0 0 -2√-2 0 0 2√-2 0 0 0 0 0 0 complex lifted from D4.3D4 ρ26 4 -4 -4 4 0 0 0 0 0 0 0 0 0 0 0 0 2√-2 0 0 -2√-2 0 0 0 0 0 0 complex lifted from D4.3D4

Smallest permutation representation of M4(2).10D4
On 32 points
Generators in S32
```(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)
(2 6)(4 8)(10 14)(12 16)(18 22)(20 24)(26 30)(28 32)
(1 11 25 17)(2 12 26 18)(3 9 27 23)(4 10 28 24)(5 15 29 21)(6 16 30 22)(7 13 31 19)(8 14 32 20)
(1 8)(2 7)(3 6)(4 5)(9 20)(10 19)(11 18)(12 17)(13 24)(14 23)(15 22)(16 21)(25 32)(26 31)(27 30)(28 29)```

`G:=sub<Sym(32)| (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), (2,6)(4,8)(10,14)(12,16)(18,22)(20,24)(26,30)(28,32), (1,11,25,17)(2,12,26,18)(3,9,27,23)(4,10,28,24)(5,15,29,21)(6,16,30,22)(7,13,31,19)(8,14,32,20), (1,8)(2,7)(3,6)(4,5)(9,20)(10,19)(11,18)(12,17)(13,24)(14,23)(15,22)(16,21)(25,32)(26,31)(27,30)(28,29)>;`

`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), (2,6)(4,8)(10,14)(12,16)(18,22)(20,24)(26,30)(28,32), (1,11,25,17)(2,12,26,18)(3,9,27,23)(4,10,28,24)(5,15,29,21)(6,16,30,22)(7,13,31,19)(8,14,32,20), (1,8)(2,7)(3,6)(4,5)(9,20)(10,19)(11,18)(12,17)(13,24)(14,23)(15,22)(16,21)(25,32)(26,31)(27,30)(28,29) );`

`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)], [(2,6),(4,8),(10,14),(12,16),(18,22),(20,24),(26,30),(28,32)], [(1,11,25,17),(2,12,26,18),(3,9,27,23),(4,10,28,24),(5,15,29,21),(6,16,30,22),(7,13,31,19),(8,14,32,20)], [(1,8),(2,7),(3,6),(4,5),(9,20),(10,19),(11,18),(12,17),(13,24),(14,23),(15,22),(16,21),(25,32),(26,31),(27,30),(28,29)]])`

Matrix representation of M4(2).10D4 in GL6(𝔽17)

 13 0 0 0 0 0 4 4 0 0 0 0 0 0 0 0 11 11 0 0 0 0 3 0 0 0 11 11 0 0 0 0 3 0 0 0
,
 16 0 0 0 0 0 0 16 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 16 0 0 0 0 0 0 16
,
 1 2 0 0 0 0 16 16 0 0 0 0 0 0 11 11 0 0 0 0 3 6 0 0 0 0 0 0 0 6 0 0 0 0 3 0
,
 4 8 0 0 0 0 13 13 0 0 0 0 0 0 0 0 11 11 0 0 0 0 3 6 0 0 11 11 0 0 0 0 3 6 0 0

`G:=sub<GL(6,GF(17))| [13,4,0,0,0,0,0,4,0,0,0,0,0,0,0,0,11,3,0,0,0,0,11,0,0,0,11,3,0,0,0,0,11,0,0,0],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[1,16,0,0,0,0,2,16,0,0,0,0,0,0,11,3,0,0,0,0,11,6,0,0,0,0,0,0,0,3,0,0,0,0,6,0],[4,13,0,0,0,0,8,13,0,0,0,0,0,0,0,0,11,3,0,0,0,0,11,6,0,0,11,3,0,0,0,0,11,6,0,0] >;`

M4(2).10D4 in GAP, Magma, Sage, TeX

`M_4(2)._{10}D_4`
`% in TeX`

`G:=Group("M4(2).10D4");`
`// GroupNames label`

`G:=SmallGroup(128,783);`
`// by ID`

`G=gap.SmallGroup(128,783);`
`# by ID`

`G:=PCGroup([7,-2,2,2,-2,2,2,-2,141,422,387,58,2019,1018,248,2804,1411,172,4037,1027]);`
`// Polycyclic`

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

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