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## G = M4(2).25C23order 128 = 27

### 7th non-split extension by M4(2) of C23 acting via C23/C22=C2

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C22 — M4(2).25C23
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C2×C4○D4 — C2×2- 1+4 — M4(2).25C23
 Lower central C1 — C2 — C22 — M4(2).25C23
 Upper central C1 — C2 — C2×C4○D4 — M4(2).25C23
 Jennings C1 — C2 — C2 — C2×C4 — M4(2).25C23

Generators and relations for M4(2).25C23
G = < a,b,c,d,e | a8=b2=d2=e2=1, c2=a2b, bab=a5, cac-1=ab, ad=da, ae=ea, cbc-1=a4b, bd=db, be=eb, cd=dc, ce=ec, ede=a4d >

Subgroups: 548 in 352 conjugacy classes, 170 normal (9 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C2×C8, M4(2), M4(2), C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C2×Q8, C4○D4, C4○D4, C4.D4, C4.10D4, C2×M4(2), C8○D4, C22×Q8, C2×C4○D4, C2×C4○D4, 2- 1+4, C2×C4.10D4, M4(2).8C22, Q8○M4(2), C2×2- 1+4, M4(2).25C23
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, C22×C4, C2×D4, C24, C2×C22⋊C4, C23×C4, C22×D4, C22×C22⋊C4, M4(2).25C23

Smallest permutation representation of M4(2).25C23
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)
(2 6)(4 8)(9 13)(11 15)(18 22)(20 24)(26 30)(28 32)
(1 15 3 13 5 11 7 9)(2 12 8 14 6 16 4 10)(17 26 19 32 21 30 23 28)(18 31 24 25 22 27 20 29)
(1 23)(2 24)(3 17)(4 18)(5 19)(6 20)(7 21)(8 22)(9 30)(10 31)(11 32)(12 25)(13 26)(14 27)(15 28)(16 29)
(1 31)(2 32)(3 25)(4 26)(5 27)(6 28)(7 29)(8 30)(9 18)(10 19)(11 20)(12 21)(13 22)(14 23)(15 24)(16 17)```

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

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

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

41 conjugacy classes

 class 1 2A 2B ··· 2H 2I 2J 4A ··· 4H 4I ··· 4N 8A ··· 8P order 1 2 2 ··· 2 2 2 4 ··· 4 4 ··· 4 8 ··· 8 size 1 1 2 ··· 2 4 4 2 ··· 2 4 ··· 4 4 ··· 4

41 irreducible representations

 dim 1 1 1 1 1 1 1 2 8 type + + + + + + - image C1 C2 C2 C2 C2 C4 C4 D4 M4(2).25C23 kernel M4(2).25C23 C2×C4.10D4 M4(2).8C22 Q8○M4(2) C2×2- 1+4 C22×Q8 C2×C4○D4 C4○D4 C1 # reps 1 6 6 2 1 4 12 8 1

Matrix representation of M4(2).25C23 in GL8(𝔽17)

 2 2 8 9 0 0 0 9 2 2 9 8 0 0 9 0 8 9 2 2 0 9 0 0 9 8 2 2 9 0 0 0 0 0 0 0 15 15 9 8 0 0 0 4 15 15 8 9 0 0 0 0 9 8 15 15 0 4 0 0 8 9 15 15
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 15 2 9 9 16 0 0 0 2 15 9 9 0 16 0 0 9 9 15 2 0 0 16 0 9 9 2 15 0 0 0 16
,
 15 2 9 9 15 0 0 0 2 15 9 9 0 15 0 0 9 9 15 2 0 0 15 0 9 9 2 15 0 0 0 15 0 0 0 0 2 15 8 8 1 0 0 0 15 2 8 8 0 0 0 0 8 8 2 15 0 0 1 0 8 8 15 2
,
 0 4 0 0 0 0 0 0 13 0 0 0 0 0 0 0 0 0 0 13 0 0 0 0 0 0 4 0 0 0 0 0 9 0 0 15 0 4 0 0 0 8 2 0 13 0 0 0 0 2 8 0 0 0 0 13 15 0 0 9 0 0 4 0
,
 0 0 1 0 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 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0

`G:=sub<GL(8,GF(17))| [2,2,8,9,0,0,0,0,2,2,9,8,0,0,0,4,8,9,2,2,0,0,0,0,9,8,2,2,0,4,0,0,0,0,0,9,15,15,9,8,0,0,9,0,15,15,8,9,0,9,0,0,9,8,15,15,9,0,0,0,8,9,15,15],[1,0,0,0,15,2,9,9,0,1,0,0,2,15,9,9,0,0,1,0,9,9,15,2,0,0,0,1,9,9,2,15,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16],[15,2,9,9,0,1,0,0,2,15,9,9,0,0,0,0,9,9,15,2,0,0,0,1,9,9,2,15,0,0,0,0,15,0,0,0,2,15,8,8,0,15,0,0,15,2,8,8,0,0,15,0,8,8,2,15,0,0,0,15,8,8,15,2],[0,13,0,0,9,0,0,15,4,0,0,0,0,8,2,0,0,0,0,4,0,2,8,0,0,0,13,0,15,0,0,9,0,0,0,0,0,13,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,13,0],[0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0] >;`

M4(2).25C23 in GAP, Magma, Sage, TeX

`M_4(2)._{25}C_2^3`
`% in TeX`

`G:=Group("M4(2).25C2^3");`
`// GroupNames label`

`G:=SmallGroup(128,1621);`
`// by ID`

`G=gap.SmallGroup(128,1621);`
`# by ID`

`G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,456,521,2804,2028,124]);`
`// Polycyclic`

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

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