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## G = M4(2).24D4order 128 = 27

### 5th 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).24D4
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C42⋊C2 — C8○2M4(2) — M4(2).24D4
 Lower central C1 — C2 — C2×C4 — M4(2).24D4
 Upper central C1 — C2×C4 — C22×C8 — M4(2).24D4
 Jennings C1 — C2 — C2 — C22×C4 — M4(2).24D4

Generators and relations for M4(2).24D4
G = < a,b,c,d | a8=b2=d2=1, c4=a4, bab=a5, ac=ca, dad=a-1b, bc=cb, dbd=a4b, dcd=a6c3 >

Subgroups: 212 in 112 conjugacy classes, 48 normal (46 characteristic)
C1, C2 [×3], C2 [×3], C4 [×4], C4 [×5], C22 [×3], C22 [×5], C8 [×7], C2×C4 [×6], C2×C4 [×9], D4 [×4], Q8 [×2], C23, C23, C42 [×3], C22⋊C4, C4⋊C4 [×2], C2×C8 [×4], C2×C8 [×6], M4(2) [×2], M4(2) [×3], C22×C4, C22×C4 [×2], C2×D4, C2×D4, C2×Q8, C4○D4 [×4], C4×C8, C8⋊C4, C22⋊C8 [×2], D4⋊C4 [×2], Q8⋊C4 [×2], C4≀C2 [×2], C8.C4 [×2], C2×C42, C42⋊C2, C22×C8 [×2], C2×M4(2) [×2], C2×C4○D4, C22.7C42, C426C4, C82M4(2), (C22×C8)⋊C2, C23.24D4, C2×C4≀C2, C2×C8.C4, M4(2).24D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, C22×C4, C2×D4 [×2], C4○D4 [×4], C42⋊C2, C4×D4 [×2], C4⋊D4, C22.D4, C4.4D4, C422C2, C24.C22, C8○D8, C8.26D4, M4(2).24D4

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

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

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

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

38 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 4A 4B 4C 4D 4E 4F 4G ··· 4N 4O 8A 8B 8C 8D 8E ··· 8N 8O 8P order 1 2 2 2 2 2 2 4 4 4 4 4 4 4 ··· 4 4 8 8 8 8 8 ··· 8 8 8 size 1 1 1 1 2 2 8 1 1 1 1 2 2 4 ··· 4 8 2 2 2 2 4 ··· 4 8 8

38 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 4 type + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C4 C4 D4 D4 C4○D4 C4○D4 C8○D8 C8.26D4 kernel M4(2).24D4 C22.7C42 C42⋊6C4 C8○2M4(2) (C22×C8)⋊C2 C23.24D4 C2×C4≀C2 C2×C8.C4 D4⋊C4 Q8⋊C4 C2×C8 M4(2) C2×C4 C23 C2 C2 # reps 1 1 1 1 1 1 1 1 4 4 2 2 6 2 8 2

Matrix representation of M4(2).24D4 in GL4(𝔽17) generated by

 1 0 0 0 0 1 0 0 0 0 0 15 0 0 2 0
,
 1 0 0 0 0 1 0 0 0 0 16 0 0 0 0 1
,
 0 1 0 0 16 0 0 0 0 0 8 0 0 0 0 8
,
 1 0 0 0 0 16 0 0 0 0 0 15 0 0 8 0
`G:=sub<GL(4,GF(17))| [1,0,0,0,0,1,0,0,0,0,0,2,0,0,15,0],[1,0,0,0,0,1,0,0,0,0,16,0,0,0,0,1],[0,16,0,0,1,0,0,0,0,0,8,0,0,0,0,8],[1,0,0,0,0,16,0,0,0,0,0,8,0,0,15,0] >;`

M4(2).24D4 in GAP, Magma, Sage, TeX

`M_4(2)._{24}D_4`
`% in TeX`

`G:=Group("M4(2).24D4");`
`// GroupNames label`

`G:=SmallGroup(128,661);`
`// by ID`

`G=gap.SmallGroup(128,661);`
`# by ID`

`G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,288,422,58,2804,718,172,124]);`
`// Polycyclic`

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

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