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## G = M4(2).10C23order 128 = 27

### 10th non-split extension by M4(2) of C23 acting via C23/C2=C22

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C4 — M4(2).10C23
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C2×C4○D4 — C2×C8○D4 — M4(2).10C23
 Lower central C1 — C2 — C2×C4 — M4(2).10C23
 Upper central C1 — C4 — C22×C4 — M4(2).10C23
 Jennings C1 — C2 — C2 — C2×C4 — M4(2).10C23

Generators and relations for M4(2).10C23
G = < a,b,c,d,e | a8=b2=1, c2=a6b, d2=e2=a4, bab=a5, cac-1=a-1b, dad-1=a3, ae=ea, bc=cb, bd=db, be=eb, dcd-1=a6bc, ce=ec, de=ed >

Subgroups: 412 in 226 conjugacy classes, 98 normal (38 characteristic)
C1, C2, C2 [×7], C4 [×4], C4 [×4], C22 [×3], C22 [×9], C8 [×4], C8 [×6], C2×C4 [×6], C2×C4 [×11], D4 [×2], D4 [×13], Q8 [×2], Q8 [×5], C23, C23 [×3], C2×C8 [×6], C2×C8 [×7], M4(2) [×6], M4(2) [×7], D8 [×6], SD16 [×12], Q16 [×6], C22×C4, C22×C4 [×3], C2×D4, C2×D4 [×2], C2×D4 [×3], C2×Q8, C2×Q8 [×2], C4○D4 [×4], C4○D4 [×10], C4.D4 [×4], C4.10D4 [×4], C8.C4 [×4], C22×C8, C22×C8, C2×M4(2), C2×M4(2) [×2], C2×M4(2), C8○D4 [×4], C8○D4 [×2], C2×D8, C2×SD16 [×2], C2×Q16, C4○D8 [×8], C8⋊C22 [×4], C8⋊C22 [×2], C8.C22 [×4], C8.C22 [×2], C2×C4○D4, C2×C4○D4 [×2], M4(2).8C22 [×2], C2×C8.C4, D4.3D4 [×4], D4.4D4 [×2], D4.5D4 [×2], C2×C8○D4, C2×C4○D8, D8⋊C22 [×2], M4(2).10C23
Quotients: C1, C2 [×15], C22 [×35], D4 [×8], C23 [×15], C2×D4 [×12], C4○D4 [×2], C24, C4⋊D4 [×4], C22×D4 [×2], C2×C4○D4, C2×C4⋊D4, M4(2).10C23

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

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

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

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

32 conjugacy classes

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

32 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 4 type + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C2 D4 D4 D4 D4 C4○D4 C4○D4 M4(2).10C23 kernel M4(2).10C23 M4(2).8C22 C2×C8.C4 D4.3D4 D4.4D4 D4.5D4 C2×C8○D4 C2×C4○D8 D8⋊C22 C2×C8 C2×D4 C2×Q8 C4○D4 C2×C4 C23 C1 # reps 1 2 1 4 2 2 1 1 2 4 1 1 2 2 2 4

Matrix representation of M4(2).10C23 in GL4(𝔽17) generated by

 0 0 12 12 0 0 12 5 5 12 0 0 12 12 0 0
,
 1 0 0 0 0 1 0 0 0 0 16 0 0 0 0 16
,
 12 5 0 0 12 12 0 0 0 0 12 5 0 0 12 12
,
 12 12 0 0 12 5 0 0 0 0 5 12 0 0 12 12
,
 13 0 0 0 0 13 0 0 0 0 13 0 0 0 0 13
`G:=sub<GL(4,GF(17))| [0,0,5,12,0,0,12,12,12,12,0,0,12,5,0,0],[1,0,0,0,0,1,0,0,0,0,16,0,0,0,0,16],[12,12,0,0,5,12,0,0,0,0,12,12,0,0,5,12],[12,12,0,0,12,5,0,0,0,0,5,12,0,0,12,12],[13,0,0,0,0,13,0,0,0,0,13,0,0,0,0,13] >;`

M4(2).10C23 in GAP, Magma, Sage, TeX

`M_4(2)._{10}C_2^3`
`% in TeX`

`G:=Group("M4(2).10C2^3");`
`// GroupNames label`

`G:=SmallGroup(128,1799);`
`// by ID`

`G=gap.SmallGroup(128,1799);`
`# by ID`

`G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,120,758,248,2804,172,4037,1027,124]);`
`// Polycyclic`

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

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