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## G = M4(2).29C23order 128 = 27

### 11st 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 — C4 — M4(2).29C23
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C2×C4○D4 — C2×C8○D4 — M4(2).29C23
 Lower central C1 — C2 — C4 — M4(2).29C23
 Upper central C1 — C4 — C2×C4○D4 — M4(2).29C23
 Jennings C1 — C2 — C2 — C2×C4 — M4(2).29C23

Generators and relations for M4(2).29C23
G = < a,b,c,d,e | a8=b2=d2=e2=1, c2=a2, bab=a5, cac-1=a-1b, ad=da, ae=ea, cbc-1=a4b, bd=db, be=eb, cd=dc, ce=ec, ede=a4d >

Subgroups: 276 in 220 conjugacy classes, 170 normal (13 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C8, C8, C8, C2×C4, C2×C4, D4, Q8, C23, C2×C8, C2×C8, C2×C8, M4(2), M4(2), C22×C4, C2×D4, C2×Q8, C4○D4, C8.C4, C22×C8, C2×M4(2), C8○D4, C8○D4, C2×C4○D4, C2×C8.C4, M4(2).C4, C2×C8○D4, Q8○M4(2), M4(2).29C23
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, C4⋊C4, C22×C4, C2×D4, C2×Q8, C24, C2×C4⋊C4, C23×C4, C22×D4, C22×Q8, C22×C4⋊C4, M4(2).29C23

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

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

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

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

44 conjugacy classes

 class 1 2A 2B ··· 2H 4A 4B 4C ··· 4I 8A 8B 8C 8D 8E ··· 8Z order 1 2 2 ··· 2 4 4 4 ··· 4 8 8 8 8 8 ··· 8 size 1 1 2 ··· 2 1 1 2 ··· 2 2 2 2 2 4 ··· 4

44 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 4 type + + + + + - - + image C1 C2 C2 C2 C2 C4 Q8 Q8 D4 M4(2).29C23 kernel M4(2).29C23 C2×C8.C4 M4(2).C4 C2×C8○D4 Q8○M4(2) C8○D4 C2×D4 C2×Q8 C4○D4 C1 # reps 1 6 6 1 2 16 3 1 4 4

Matrix representation of M4(2).29C23 in GL4(𝔽17) generated by

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

M4(2).29C23 in GAP, Magma, Sage, TeX

`M_4(2)._{29}C_2^3`
`% in TeX`

`G:=Group("M4(2).29C2^3");`
`// GroupNames label`

`G:=SmallGroup(128,1648);`
`// by ID`

`G=gap.SmallGroup(128,1648);`
`# by ID`

`G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,120,521,2804,172,124]);`
`// Polycyclic`

`G:=Group<a,b,c,d,e|a^8=b^2=d^2=e^2=1,c^2=a^2,b*a*b=a^5,c*a*c^-1=a^-1*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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