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## G = M4(2).C23order 128 = 27

### 4th non-split extension by M4(2) of C23 acting via C23/C2=C22

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C4 — M4(2).C23
 Chief series C1 — C2 — C22 — C2×C4 — C22×C4 — C22×Q8 — C2×2- 1+4 — M4(2).C23
 Lower central C1 — C2 — C2×C4 — M4(2).C23
 Upper central C1 — C2 — C22×C4 — M4(2).C23
 Jennings C1 — C2 — C2 — C2×C4 — M4(2).C23

Generators and relations for M4(2).C23
G = < a,b,c,d,e | a8=b2=c2=1, d2=a4, e2=a6, bab=a5, cac=a3, dad-1=a5b, ae=ea, cbc=ebe-1=a4b, bd=db, dcd-1=a4c, ece-1=a6c, ede-1=a4bd >

Subgroups: 636 in 349 conjugacy classes, 106 normal (18 characteristic)
C1, C2, C2 [×8], C4 [×4], C4 [×11], C22 [×3], C22 [×12], C8 [×4], C2×C4 [×2], C2×C4 [×8], C2×C4 [×35], D4 [×4], D4 [×22], Q8 [×4], Q8 [×20], C23, C23 [×3], C42 [×2], C22⋊C4 [×5], C4⋊C4 [×5], C2×C8 [×2], M4(2) [×4], M4(2) [×2], D8 [×4], SD16 [×8], Q16 [×4], C22×C4, C22×C4 [×2], C22×C4 [×7], C2×D4, C2×D4 [×2], C2×D4 [×5], C2×Q8, C2×Q8 [×6], C2×Q8 [×22], C4○D4 [×8], C4○D4 [×40], C4.10D4 [×4], C4≀C2 [×8], C42⋊C2, C22⋊Q8 [×2], C22.D4 [×2], C4.4D4 [×2], C4⋊Q8 [×2], C2×M4(2) [×2], C4○D8 [×4], C8⋊C22 [×4], C8⋊C22 [×2], C8.C22 [×4], C8.C22 [×2], C22×Q8, C22×Q8 [×2], C2×C4○D4, C2×C4○D4 [×2], C2×C4○D4 [×4], 2- 1+4 [×4], 2- 1+4 [×6], C2×C4.10D4, C42⋊C22 [×2], D4.8D4 [×4], D4.10D4 [×4], C23.38C23, D8⋊C22 [×2], C2×2- 1+4, M4(2).C23
Quotients: C1, C2 [×15], C22 [×35], D4 [×12], C23 [×15], C2×D4 [×18], C24, C22≀C2 [×4], C22×D4 [×3], C2×C22≀C2, M4(2).C23

Character table of M4(2).C23

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 4M 4N 4O 8A 8B 8C 8D size 1 1 2 2 2 4 4 4 4 8 2 2 2 2 4 4 4 4 4 4 4 4 8 8 8 8 8 8 8 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 1 1 1 -1 -1 -1 -1 -1 1 1 1 1 1 1 1 -1 -1 -1 -1 1 -1 -1 -1 1 1 1 1 linear of order 2 ρ3 1 1 -1 1 -1 1 -1 -1 1 1 -1 -1 1 1 -1 -1 1 1 -1 -1 1 1 1 -1 -1 1 1 -1 -1 linear of order 2 ρ4 1 1 -1 1 -1 -1 1 1 -1 -1 -1 -1 1 1 -1 -1 1 -1 1 1 -1 1 -1 1 1 1 1 -1 -1 linear of order 2 ρ5 1 1 -1 1 -1 1 1 -1 -1 1 -1 -1 1 1 1 1 -1 -1 1 -1 1 -1 -1 -1 1 1 -1 1 -1 linear of order 2 ρ6 1 1 -1 1 -1 -1 -1 1 1 -1 -1 -1 1 1 1 1 -1 1 -1 1 -1 -1 1 1 -1 1 -1 1 -1 linear of order 2 ρ7 1 1 1 1 1 1 -1 1 -1 1 1 1 1 1 -1 -1 -1 -1 -1 1 1 -1 -1 1 -1 1 -1 -1 1 linear of order 2 ρ8 1 1 1 1 1 -1 1 -1 1 -1 1 1 1 1 -1 -1 -1 1 1 -1 -1 -1 1 -1 1 1 -1 -1 1 linear of order 2 ρ9 1 1 1 1 1 1 1 1 1 -1 1 1 1 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 linear of order 2 ρ10 1 1 1 1 1 -1 -1 -1 -1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 1 1 1 1 -1 -1 -1 -1 linear of order 2 ρ11 1 1 -1 1 -1 1 -1 -1 1 -1 -1 -1 1 1 -1 -1 1 1 -1 -1 1 1 -1 1 1 -1 -1 1 1 linear of order 2 ρ12 1 1 -1 1 -1 -1 1 1 -1 1 -1 -1 1 1 -1 -1 1 -1 1 1 -1 1 1 -1 -1 -1 -1 1 1 linear of order 2 ρ13 1 1 -1 1 -1 1 1 -1 -1 -1 -1 -1 1 1 1 1 -1 -1 1 -1 1 -1 1 1 -1 -1 1 -1 1 linear of order 2 ρ14 1 1 -1 1 -1 -1 -1 1 1 1 -1 -1 1 1 1 1 -1 1 -1 1 -1 -1 -1 -1 1 -1 1 -1 1 linear of order 2 ρ15 1 1 1 1 1 1 -1 1 -1 -1 1 1 1 1 -1 -1 -1 -1 -1 1 1 -1 1 -1 1 -1 1 1 -1 linear of order 2 ρ16 1 1 1 1 1 -1 1 -1 1 1 1 1 1 1 -1 -1 -1 1 1 -1 -1 -1 -1 1 -1 -1 1 1 -1 linear of order 2 ρ17 2 2 2 -2 -2 2 0 2 0 0 -2 2 -2 2 0 0 0 0 0 -2 -2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ18 2 2 -2 2 -2 0 0 0 0 0 2 2 -2 -2 2 -2 2 0 0 0 0 -2 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ19 2 2 2 -2 -2 0 2 0 2 0 2 -2 2 -2 0 0 0 -2 -2 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ20 2 2 -2 -2 2 -2 0 2 0 0 2 -2 -2 2 0 0 0 0 0 -2 2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ21 2 2 2 2 2 0 0 0 0 0 -2 -2 -2 -2 2 -2 -2 0 0 0 0 2 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ22 2 2 -2 -2 2 0 2 0 -2 0 -2 2 2 -2 0 0 0 2 -2 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ23 2 2 2 -2 -2 -2 0 -2 0 0 -2 2 -2 2 0 0 0 0 0 2 2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ24 2 2 2 -2 -2 0 -2 0 -2 0 2 -2 2 -2 0 0 0 2 2 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ25 2 2 2 2 2 0 0 0 0 0 -2 -2 -2 -2 -2 2 2 0 0 0 0 -2 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ26 2 2 -2 2 -2 0 0 0 0 0 2 2 -2 -2 -2 2 -2 0 0 0 0 2 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ27 2 2 -2 -2 2 2 0 -2 0 0 2 -2 -2 2 0 0 0 0 0 2 -2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ28 2 2 -2 -2 2 0 -2 0 2 0 -2 2 2 -2 0 0 0 -2 2 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ29 8 -8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 symplectic faithful, Schur index 2

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

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

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

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

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

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

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

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

`M_4(2).C_2^3`
`% in TeX`

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

`G:=SmallGroup(128,1752);`
`// by ID`

`G=gap.SmallGroup(128,1752);`
`# by ID`

`G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,758,352,2019,2804,1411,718,172,2028]);`
`// Polycyclic`

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

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