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## G = M4(2).45D4order 128 = 27

### 9th non-split extension by M4(2) of D4 acting via D4/C22=C2

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C23 — M4(2).45D4
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C23×C4 — C22×M4(2) — M4(2).45D4
 Lower central C1 — C2 — C23 — M4(2).45D4
 Upper central C1 — C22 — C23×C4 — M4(2).45D4
 Jennings C1 — C2 — C2 — C22×C4 — M4(2).45D4

Generators and relations for M4(2).45D4
G = < a,b,c,d | a8=b2=1, c4=a4, d2=a2b, bab=a5, cac-1=dad-1=ab, cbc-1=dbd-1=a4b, dcd-1=a6bc3 >

Subgroups: 316 in 169 conjugacy classes, 58 normal (28 characteristic)
C1, C2 [×3], C2 [×5], C4 [×2], C4 [×2], C4 [×6], C22, C22 [×4], C22 [×11], C8 [×6], C2×C4 [×4], C2×C4 [×4], C2×C4 [×20], Q8 [×4], C23 [×3], C23 [×5], C22⋊C4 [×4], C4⋊C4 [×6], C2×C8 [×8], M4(2) [×4], M4(2) [×10], C22×C4 [×6], C22×C4 [×4], C22×C4 [×4], C2×Q8 [×6], C24, C22⋊C8 [×2], C4.10D4 [×4], C2×C22⋊C4 [×2], C2×C4⋊C4, C2×C4⋊C4 [×2], C22⋊Q8 [×4], C22×C8, C2×M4(2) [×4], C2×M4(2) [×5], C23×C4, C22×Q8, C22.C42 [×2], C24.4C4, C2×C4.10D4 [×2], C2×C22⋊Q8, C22×M4(2), M4(2).45D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×8], C23, C22⋊C4 [×4], C22×C4, C2×D4 [×4], C4○D4 [×2], C4.10D4 [×2], C2×C22⋊C4, C4×D4 [×2], C22≀C2, C4⋊D4 [×2], C22.D4, C23.23D4, C2×C4.10D4, M4(2).8C22, M4(2).45D4

Smallest permutation representation of M4(2).45D4
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 22)(2 19)(3 24)(4 21)(5 18)(6 23)(7 20)(8 17)(9 30)(10 27)(11 32)(12 29)(13 26)(14 31)(15 28)(16 25)
(1 13 3 11 5 9 7 15)(2 27 8 29 6 31 4 25)(10 21 12 19 14 17 16 23)(18 26 20 32 22 30 24 28)
(1 26 24 15 5 30 20 11)(2 14 21 25 6 10 17 29)(3 32 18 13 7 28 22 9)(4 12 23 31 8 16 19 27)```

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

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

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

32 conjugacy classes

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

32 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 4 4 type + + + + + + + + - image C1 C2 C2 C2 C2 C2 C4 C4 D4 D4 C4○D4 C4.10D4 M4(2).8C22 kernel M4(2).45D4 C22.C42 C24.4C4 C2×C4.10D4 C2×C22⋊Q8 C22×M4(2) C2×C22⋊C4 C2×C4⋊C4 M4(2) C22×C4 C2×C4 C22 C2 # reps 1 2 1 2 1 1 4 4 4 4 4 2 2

Matrix representation of M4(2).45D4 in GL6(𝔽17)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 16 15 0 0 0 0 1 1 0 0
,
 1 0 0 0 0 0 0 1 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
,
 11 6 0 0 0 0 8 6 0 0 0 0 0 0 0 0 7 7 0 0 0 0 5 10 0 0 0 7 0 0 0 0 12 0 0 0
,
 6 15 0 0 0 0 9 11 0 0 0 0 0 0 0 0 10 10 0 0 0 0 12 7 0 0 0 10 0 0 0 0 5 0 0 0

`G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,1,0,0,0,0,15,1,0,0,1,0,0,0,0,0,0,1,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[11,8,0,0,0,0,6,6,0,0,0,0,0,0,0,0,0,12,0,0,0,0,7,0,0,0,7,5,0,0,0,0,7,10,0,0],[6,9,0,0,0,0,15,11,0,0,0,0,0,0,0,0,0,5,0,0,0,0,10,0,0,0,10,12,0,0,0,0,10,7,0,0] >;`

M4(2).45D4 in GAP, Magma, Sage, TeX

`M_4(2)._{45}D_4`
`% in TeX`

`G:=Group("M4(2).45D4");`
`// GroupNames label`

`G:=SmallGroup(128,633);`
`// by ID`

`G=gap.SmallGroup(128,633);`
`# by ID`

`G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,736,422,2019,1018,2028,124]);`
`// Polycyclic`

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

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