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## G = M4(2).11D4order 128 = 27

### 11st 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).11D4
 Chief series C1 — C2 — C22 — C2×C4 — C22×C4 — C2×M4(2) — C2×C4.10D4 — M4(2).11D4
 Lower central C1 — C2 — C22×C4 — M4(2).11D4
 Upper central C1 — C22 — C22×C4 — M4(2).11D4
 Jennings C1 — C2 — C2 — C22×C4 — M4(2).11D4

Generators and relations for M4(2).11D4
G = < a,b,c,d | a8=b2=c4=d2=1, bab=a5, cac-1=ab, dad=a3, bc=cb, dbd=a4b, dcd=a6c-1 >

Subgroups: 272 in 131 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), SD16, Q16, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C2×Q8, C4○D4, C22⋊C8, C4.10D4, D4⋊C4, Q8⋊C4, C8.C4, C2×C4⋊C4, C22×C8, C2×M4(2), C2×SD16, C2×Q16, C8.C22, C22×Q8, C2×C4○D4, C22.C42, (C22×C8)⋊C2, C2×C4.10D4, C2×Q8⋊C4, C23.36D4, C2×C8.C4, C2×C8.C22, M4(2).11D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C22≀C2, C4⋊D4, C22.D4, C4.4D4, C23.10D4, D4.3D4, D4.5D4, M4(2).11D4

Character table of M4(2).11D4

 class 1 2A 2B 2C 2D 2E 2F 4A 4B 4C 4D 4E 4F 4G 4H 4I 8A 8B 8C 8D 8E 8F 8G 8H 8I 8J size 1 1 1 1 2 2 8 2 2 2 2 8 8 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 0 -2 2 -2 2 0 0 0 0 0 0 0 0 0 -2 0 0 2 0 0 orthogonal lifted from D4 ρ10 2 -2 2 -2 2 -2 0 -2 2 -2 2 0 0 0 0 0 0 0 0 0 2 0 0 -2 0 0 orthogonal lifted from D4 ρ11 2 2 2 2 -2 -2 2 -2 2 2 -2 0 0 -2 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ12 2 -2 2 -2 -2 2 0 2 2 -2 -2 0 -2 0 0 2 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ13 2 2 2 2 -2 -2 0 2 -2 -2 2 0 0 0 0 0 -2 2 2 -2 0 0 0 0 0 0 orthogonal lifted from D4 ρ14 2 -2 2 -2 -2 2 0 2 2 -2 -2 0 2 0 0 -2 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ15 2 2 2 2 -2 -2 0 2 -2 -2 2 0 0 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 -2 2 2 -2 0 0 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 -2 -2 2 2 0 0 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 -2 -2 2 2 0 0 0 0 0 0 0 0 0 0 2i 0 0 0 -2i complex lifted from C4○D4 ρ19 2 2 2 2 2 2 0 -2 -2 -2 -2 0 0 0 0 0 0 0 0 0 0 0 2i 0 -2i 0 complex lifted from C4○D4 ρ20 2 -2 2 -2 2 -2 0 2 -2 2 -2 2i 0 0 -2i 0 0 0 0 0 0 0 0 0 0 0 complex lifted from C4○D4 ρ21 2 2 2 2 2 2 0 -2 -2 -2 -2 0 0 0 0 0 0 0 0 0 0 0 -2i 0 2i 0 complex lifted from C4○D4 ρ22 2 -2 2 -2 2 -2 0 2 -2 2 -2 -2i 0 0 2i 0 0 0 0 0 0 0 0 0 0 0 complex lifted from C4○D4 ρ23 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 symplectic lifted from D4.5D4, Schur index 2 ρ24 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 symplectic lifted from D4.5D4, Schur index 2 ρ25 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 complex lifted from D4.3D4 ρ26 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 complex lifted from D4.3D4

Smallest permutation representation of M4(2).11D4
On 64 points
Generators in S64
```(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)(49 50 51 52 53 54 55 56)(57 58 59 60 61 62 63 64)
(1 27)(2 32)(3 29)(4 26)(5 31)(6 28)(7 25)(8 30)(9 64)(10 61)(11 58)(12 63)(13 60)(14 57)(15 62)(16 59)(17 50)(18 55)(19 52)(20 49)(21 54)(22 51)(23 56)(24 53)(33 45)(34 42)(35 47)(36 44)(37 41)(38 46)(39 43)(40 48)
(1 57 22 43)(2 11 23 36)(3 63 24 41)(4 9 17 34)(5 61 18 47)(6 15 19 40)(7 59 20 45)(8 13 21 38)(10 55 35 31)(12 53 37 29)(14 51 39 27)(16 49 33 25)(26 64 50 42)(28 62 52 48)(30 60 54 46)(32 58 56 44)
(2 4)(3 7)(6 8)(9 38)(10 33)(11 36)(12 39)(13 34)(14 37)(15 40)(16 35)(17 23)(19 21)(20 24)(26 28)(27 31)(30 32)(41 61)(42 64)(43 59)(44 62)(45 57)(46 60)(47 63)(48 58)(50 52)(51 55)(54 56)```

`G:=sub<Sym(64)| (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)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64), (1,27)(2,32)(3,29)(4,26)(5,31)(6,28)(7,25)(8,30)(9,64)(10,61)(11,58)(12,63)(13,60)(14,57)(15,62)(16,59)(17,50)(18,55)(19,52)(20,49)(21,54)(22,51)(23,56)(24,53)(33,45)(34,42)(35,47)(36,44)(37,41)(38,46)(39,43)(40,48), (1,57,22,43)(2,11,23,36)(3,63,24,41)(4,9,17,34)(5,61,18,47)(6,15,19,40)(7,59,20,45)(8,13,21,38)(10,55,35,31)(12,53,37,29)(14,51,39,27)(16,49,33,25)(26,64,50,42)(28,62,52,48)(30,60,54,46)(32,58,56,44), (2,4)(3,7)(6,8)(9,38)(10,33)(11,36)(12,39)(13,34)(14,37)(15,40)(16,35)(17,23)(19,21)(20,24)(26,28)(27,31)(30,32)(41,61)(42,64)(43,59)(44,62)(45,57)(46,60)(47,63)(48,58)(50,52)(51,55)(54,56)>;`

`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)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64), (1,27)(2,32)(3,29)(4,26)(5,31)(6,28)(7,25)(8,30)(9,64)(10,61)(11,58)(12,63)(13,60)(14,57)(15,62)(16,59)(17,50)(18,55)(19,52)(20,49)(21,54)(22,51)(23,56)(24,53)(33,45)(34,42)(35,47)(36,44)(37,41)(38,46)(39,43)(40,48), (1,57,22,43)(2,11,23,36)(3,63,24,41)(4,9,17,34)(5,61,18,47)(6,15,19,40)(7,59,20,45)(8,13,21,38)(10,55,35,31)(12,53,37,29)(14,51,39,27)(16,49,33,25)(26,64,50,42)(28,62,52,48)(30,60,54,46)(32,58,56,44), (2,4)(3,7)(6,8)(9,38)(10,33)(11,36)(12,39)(13,34)(14,37)(15,40)(16,35)(17,23)(19,21)(20,24)(26,28)(27,31)(30,32)(41,61)(42,64)(43,59)(44,62)(45,57)(46,60)(47,63)(48,58)(50,52)(51,55)(54,56) );`

`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),(49,50,51,52,53,54,55,56),(57,58,59,60,61,62,63,64)], [(1,27),(2,32),(3,29),(4,26),(5,31),(6,28),(7,25),(8,30),(9,64),(10,61),(11,58),(12,63),(13,60),(14,57),(15,62),(16,59),(17,50),(18,55),(19,52),(20,49),(21,54),(22,51),(23,56),(24,53),(33,45),(34,42),(35,47),(36,44),(37,41),(38,46),(39,43),(40,48)], [(1,57,22,43),(2,11,23,36),(3,63,24,41),(4,9,17,34),(5,61,18,47),(6,15,19,40),(7,59,20,45),(8,13,21,38),(10,55,35,31),(12,53,37,29),(14,51,39,27),(16,49,33,25),(26,64,50,42),(28,62,52,48),(30,60,54,46),(32,58,56,44)], [(2,4),(3,7),(6,8),(9,38),(10,33),(11,36),(12,39),(13,34),(14,37),(15,40),(16,35),(17,23),(19,21),(20,24),(26,28),(27,31),(30,32),(41,61),(42,64),(43,59),(44,62),(45,57),(46,60),(47,63),(48,58),(50,52),(51,55),(54,56)]])`

Matrix representation of M4(2).11D4 in GL6(𝔽17)

 16 15 0 0 0 0 1 1 0 0 0 0 0 0 4 0 4 13 0 0 0 13 4 13 0 0 15 2 0 4 0 0 15 2 13 0
,
 16 0 0 0 0 0 0 16 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0
,
 4 0 0 0 0 0 13 13 0 0 0 0 0 0 0 6 0 6 0 0 6 0 6 0 0 0 0 14 0 11 0 0 14 0 11 0
,
 1 0 0 0 0 0 16 16 0 0 0 0 0 0 16 0 0 0 0 0 0 1 0 0 0 0 1 0 1 0 0 0 0 16 0 16

`G:=sub<GL(6,GF(17))| [16,1,0,0,0,0,15,1,0,0,0,0,0,0,4,0,15,15,0,0,0,13,2,2,0,0,4,4,0,13,0,0,13,13,4,0],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[4,13,0,0,0,0,0,13,0,0,0,0,0,0,0,6,0,14,0,0,6,0,14,0,0,0,0,6,0,11,0,0,6,0,11,0],[1,16,0,0,0,0,0,16,0,0,0,0,0,0,16,0,1,0,0,0,0,1,0,16,0,0,0,0,1,0,0,0,0,0,0,16] >;`

M4(2).11D4 in GAP, Magma, Sage, TeX

`M_4(2)._{11}D_4`
`% in TeX`

`G:=Group("M4(2).11D4");`
`// GroupNames label`

`G:=SmallGroup(128,784);`
`// by ID`

`G=gap.SmallGroup(128,784);`
`# by ID`

`G:=PCGroup([7,-2,2,2,-2,2,2,-2,141,456,422,387,58,2019,1018,248,2804,1411,172,4037,1027]);`
`// 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*b,d*a*d=a^3,b*c=c*b,d*b*d=a^4*b,d*c*d=a^6*c^-1>;`
`// generators/relations`

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