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## G = M4(2).20D4order 128 = 27

### 1st non-split extension by M4(2) of D4 acting via D4/C4=C2

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C4 — M4(2).20D4
 Chief series C1 — C2 — C22 — C2×C4 — C22×C4 — C42⋊C2 — C8○2M4(2) — M4(2).20D4
 Lower central C1 — C2 — C2×C4 — M4(2).20D4
 Upper central C1 — C22 — C42⋊C2 — M4(2).20D4
 Jennings C1 — C2 — C2 — C2×C4 — M4(2).20D4

Generators and relations for M4(2).20D4
G = < a,b,c,d | a8=b2=1, c4=d2=a4, bab=a5, ac=ca, dad-1=a3, bc=cb, bd=db, dcd-1=a4c3 >

Subgroups: 492 in 266 conjugacy classes, 108 normal (20 characteristic)
C1, C2, C2 [×2], C2 [×4], C4 [×2], C4 [×2], C4 [×10], C22, C22 [×2], C22 [×8], C8 [×8], C2×C4 [×2], C2×C4 [×8], C2×C4 [×14], D4 [×8], Q8 [×16], C23, C23 [×2], C42 [×2], C22⋊C4 [×2], C22⋊C4 [×8], C4⋊C4 [×2], C4⋊C4 [×8], C2×C8 [×2], C2×C8 [×6], M4(2) [×4], D8 [×2], SD16 [×12], Q16 [×18], C22×C4, C22×C4 [×4], C2×D4 [×2], C2×D4 [×2], C2×Q8 [×6], C2×Q8 [×10], C4○D4 [×8], C4×C8 [×2], C8⋊C4 [×2], C42⋊C2, C22⋊Q8 [×4], C22.D4 [×4], C4.4D4 [×4], C4⋊Q8 [×4], C22×C8, C2×M4(2), C2×D8, C2×SD16 [×6], C2×Q16, C2×Q16 [×8], C2×Q16 [×4], C4○D8 [×4], C8.C22 [×8], C22×Q8 [×2], C2×C4○D4 [×2], C82M4(2), C4⋊Q16 [×2], C8.12D4 [×2], C8.2D4 [×4], C23.38C23 [×2], C22×Q16, C2×C4○D8, C2×C8.C22 [×2], M4(2).20D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×12], C23 [×15], C2×D4 [×18], C24, C41D4 [×4], C22×D4 [×3], C2×C41D4, Q8○D8 [×2], M4(2).20D4

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

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

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

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

32 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 4C 4D 4E 4F 4G 4H 4I ··· 4N 8A 8B 8C 8D 8E ··· 8J order 1 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 4 ··· 4 8 8 8 8 8 ··· 8 size 1 1 1 1 2 2 8 8 2 2 2 2 4 4 4 4 8 ··· 8 2 2 2 2 4 ··· 4

32 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 4 type + + + + + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C2 C2 C2 D4 D4 D4 D4 Q8○D8 kernel M4(2).20D4 C8○2M4(2) C4⋊Q16 C8.12D4 C8.2D4 C23.38C23 C22×Q16 C2×C4○D8 C2×C8.C22 C22⋊C4 C4⋊C4 C2×C8 M4(2) C2 # reps 1 1 2 2 4 2 1 1 2 2 2 4 4 4

Matrix representation of M4(2).20D4 in GL6(𝔽17)

 16 0 0 0 0 0 0 16 0 0 0 0 0 0 5 5 0 0 0 0 12 5 0 0 0 0 0 0 12 12 0 0 0 0 5 12
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 16 0 0 16 0 0 0 0 0 0 16 0 0
,
 0 16 0 0 0 0 1 0 0 0 0 0 0 0 3 14 0 0 0 0 3 3 0 0 0 0 0 0 3 14 0 0 0 0 3 3
,
 1 0 0 0 0 0 0 16 0 0 0 0 0 0 12 12 0 0 0 0 12 5 0 0 0 0 0 0 12 12 0 0 0 0 12 5

`G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,5,12,0,0,0,0,5,5,0,0,0,0,0,0,12,5,0,0,0,0,12,12],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,16,0,0,0,0,0,0,16,0,0],[0,1,0,0,0,0,16,0,0,0,0,0,0,0,3,3,0,0,0,0,14,3,0,0,0,0,0,0,3,3,0,0,0,0,14,3],[1,0,0,0,0,0,0,16,0,0,0,0,0,0,12,12,0,0,0,0,12,5,0,0,0,0,0,0,12,12,0,0,0,0,12,5] >;`

M4(2).20D4 in GAP, Magma, Sage, TeX

`M_4(2)._{20}D_4`
`% in TeX`

`G:=Group("M4(2).20D4");`
`// GroupNames label`

`G:=SmallGroup(128,1888);`
`// by ID`

`G=gap.SmallGroup(128,1888);`
`# by ID`

`G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,568,758,184,521,2804,172]);`
`// Polycyclic`

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

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