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## G = M4(2)⋊7Q8order 128 = 27

### 5th semidirect product of M4(2) and Q8 acting via Q8/C4=C2

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C4 — M4(2)⋊7Q8
 Chief series C1 — C2 — C22 — C23 — C22×C4 — C2×C42 — C4×M4(2) — M4(2)⋊7Q8
 Lower central C1 — C2 — C2×C4 — M4(2)⋊7Q8
 Upper central C1 — C2×C4 — C2×C42 — M4(2)⋊7Q8
 Jennings C1 — C2 — C2 — C22×C4 — M4(2)⋊7Q8

Generators and relations for M4(2)⋊7Q8
G = < a,b,c,d | a8=b2=c4=1, d2=c2, bab=a5, ac=ca, dad-1=a-1b, bc=cb, dbd-1=a4b, dcd-1=c-1 >

Subgroups: 236 in 130 conjugacy classes, 58 normal (28 characteristic)
C1, C2 [×3], C2 [×2], C4 [×6], C4 [×11], C22 [×3], C22 [×2], C8 [×4], C2×C4 [×6], C2×C4 [×4], C2×C4 [×15], Q8 [×4], C23, C42 [×4], C42 [×6], C22⋊C4 [×2], C4⋊C4 [×4], C4⋊C4 [×10], C2×C8 [×2], M4(2) [×4], M4(2) [×2], C22×C4 [×3], C22×C4 [×2], C2×Q8 [×2], C2.C42, C4×C8, C8⋊C4, C2×C42, C2×C42 [×2], C2×C4⋊C4, C42⋊C2 [×2], C4×Q8 [×2], C22⋊Q8 [×2], C42.C2 [×2], C4⋊Q8 [×2], C2×M4(2) [×2], C426C4 [×4], C4×C4⋊C4, C4×M4(2), C23.37C23, M4(2)⋊7Q8
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], Q8 [×4], C23, C22⋊C4 [×4], C22×C4, C2×D4 [×2], C2×Q8 [×2], C4○D4 [×2], C4≀C2 [×2], C2×C22⋊C4, C4×Q8 [×2], C22⋊Q8 [×2], C4.4D4, C4⋊Q8, C23.67C23, C2×C4≀C2, C42⋊C22, M4(2)⋊7Q8

Smallest permutation representation of M4(2)⋊7Q8
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 27)(2 32)(3 29)(4 26)(5 31)(6 28)(7 25)(8 30)(9 21)(10 18)(11 23)(12 20)(13 17)(14 22)(15 19)(16 24)
(1 24 27 16)(2 17 28 9)(3 18 29 10)(4 19 30 11)(5 20 31 12)(6 21 32 13)(7 22 25 14)(8 23 26 15)
(1 17 27 9)(2 16 28 24)(3 19 29 11)(4 10 30 18)(5 21 31 13)(6 12 32 20)(7 23 25 15)(8 14 26 22)```

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

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

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

38 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E ··· 4J 4K ··· 4T 4U 4V 4W 4X 8A ··· 8H order 1 2 2 2 2 2 4 4 4 4 4 ··· 4 4 ··· 4 4 4 4 4 8 ··· 8 size 1 1 1 1 2 2 1 1 1 1 2 ··· 2 4 ··· 4 8 8 8 8 4 ··· 4

38 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 4 type + + + + + + - + image C1 C2 C2 C2 C2 C4 C4 D4 Q8 D4 C4○D4 C4≀C2 C42⋊C22 kernel M4(2)⋊7Q8 C42⋊6C4 C4×C4⋊C4 C4×M4(2) C23.37C23 C42.C2 C4⋊Q8 C42 M4(2) C22×C4 C2×C4 C4 C2 # reps 1 4 1 1 1 4 4 2 4 2 4 8 2

Matrix representation of M4(2)⋊7Q8 in GL4(𝔽17) generated by

 0 1 0 0 13 0 0 0 0 0 13 8 0 0 13 4
,
 1 0 0 0 0 16 0 0 0 0 16 0 0 0 0 16
,
 16 0 0 0 0 16 0 0 0 0 1 15 0 0 1 16
,
 0 13 0 0 4 0 0 0 0 0 4 0 0 0 4 13
`G:=sub<GL(4,GF(17))| [0,13,0,0,1,0,0,0,0,0,13,13,0,0,8,4],[1,0,0,0,0,16,0,0,0,0,16,0,0,0,0,16],[16,0,0,0,0,16,0,0,0,0,1,1,0,0,15,16],[0,4,0,0,13,0,0,0,0,0,4,4,0,0,0,13] >;`

M4(2)⋊7Q8 in GAP, Magma, Sage, TeX

`M_4(2)\rtimes_7Q_8`
`% in TeX`

`G:=Group("M4(2):7Q8");`
`// GroupNames label`

`G:=SmallGroup(128,718);`
`// by ID`

`G=gap.SmallGroup(128,718);`
`# by ID`

`G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,400,422,100,1018,248,2028]);`
`// Polycyclic`

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

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