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## G = C23.7M4(2)  order 128 = 27

### 3rd non-split extension by C23 of M4(2) acting via M4(2)/C4=C22

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C4 — C23.7M4(2)
 Chief series C1 — C2 — C4 — C2×C4 — C2×C8 — C22×C8 — C42.6C22 — C23.7M4(2)
 Lower central C1 — C22 — C2×C4 — C23.7M4(2)
 Upper central C1 — C2×C4 — C22×C8 — C23.7M4(2)
 Jennings C1 — C2 — C2 — C2 — C2 — C4 — C2×C4 — C22×C8 — C23.7M4(2)

Generators and relations for C23.7M4(2)
G = < a,b,c,d,e | a2=b2=c2=1, d8=c, e2=b, ab=ba, eae-1=ac=ca, dad-1=abc, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede-1=abcd5 >

Smallest permutation representation of C23.7M4(2)
On 64 points
Generators in S64
```(2 52)(4 54)(6 56)(8 58)(10 60)(12 62)(14 64)(16 50)(17 45)(18 26)(19 47)(20 28)(21 33)(22 30)(23 35)(24 32)(25 37)(27 39)(29 41)(31 43)(34 42)(36 44)(38 46)(40 48)
(1 59)(2 60)(3 61)(4 62)(5 63)(6 64)(7 49)(8 50)(9 51)(10 52)(11 53)(12 54)(13 55)(14 56)(15 57)(16 58)(17 45)(18 46)(19 47)(20 48)(21 33)(22 34)(23 35)(24 36)(25 37)(26 38)(27 39)(28 40)(29 41)(30 42)(31 43)(32 44)
(1 9)(2 10)(3 11)(4 12)(5 13)(6 14)(7 15)(8 16)(17 25)(18 26)(19 27)(20 28)(21 29)(22 30)(23 31)(24 32)(33 41)(34 42)(35 43)(36 44)(37 45)(38 46)(39 47)(40 48)(49 57)(50 58)(51 59)(52 60)(53 61)(54 62)(55 63)(56 64)
(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)(33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48)(49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64)
(1 26 59 38)(2 43 60 31)(3 40 61 28)(4 17 62 45)(5 30 63 42)(6 47 64 19)(7 44 49 32)(8 21 50 33)(9 18 51 46)(10 35 52 23)(11 48 53 20)(12 25 54 37)(13 22 55 34)(14 39 56 27)(15 36 57 24)(16 29 58 41)```

`G:=sub<Sym(64)| (2,52)(4,54)(6,56)(8,58)(10,60)(12,62)(14,64)(16,50)(17,45)(18,26)(19,47)(20,28)(21,33)(22,30)(23,35)(24,32)(25,37)(27,39)(29,41)(31,43)(34,42)(36,44)(38,46)(40,48), (1,59)(2,60)(3,61)(4,62)(5,63)(6,64)(7,49)(8,50)(9,51)(10,52)(11,53)(12,54)(13,55)(14,56)(15,57)(16,58)(17,45)(18,46)(19,47)(20,48)(21,33)(22,34)(23,35)(24,36)(25,37)(26,38)(27,39)(28,40)(29,41)(30,42)(31,43)(32,44), (1,9)(2,10)(3,11)(4,12)(5,13)(6,14)(7,15)(8,16)(17,25)(18,26)(19,27)(20,28)(21,29)(22,30)(23,31)(24,32)(33,41)(34,42)(35,43)(36,44)(37,45)(38,46)(39,47)(40,48)(49,57)(50,58)(51,59)(52,60)(53,61)(54,62)(55,63)(56,64), (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)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64), (1,26,59,38)(2,43,60,31)(3,40,61,28)(4,17,62,45)(5,30,63,42)(6,47,64,19)(7,44,49,32)(8,21,50,33)(9,18,51,46)(10,35,52,23)(11,48,53,20)(12,25,54,37)(13,22,55,34)(14,39,56,27)(15,36,57,24)(16,29,58,41)>;`

`G:=Group( (2,52)(4,54)(6,56)(8,58)(10,60)(12,62)(14,64)(16,50)(17,45)(18,26)(19,47)(20,28)(21,33)(22,30)(23,35)(24,32)(25,37)(27,39)(29,41)(31,43)(34,42)(36,44)(38,46)(40,48), (1,59)(2,60)(3,61)(4,62)(5,63)(6,64)(7,49)(8,50)(9,51)(10,52)(11,53)(12,54)(13,55)(14,56)(15,57)(16,58)(17,45)(18,46)(19,47)(20,48)(21,33)(22,34)(23,35)(24,36)(25,37)(26,38)(27,39)(28,40)(29,41)(30,42)(31,43)(32,44), (1,9)(2,10)(3,11)(4,12)(5,13)(6,14)(7,15)(8,16)(17,25)(18,26)(19,27)(20,28)(21,29)(22,30)(23,31)(24,32)(33,41)(34,42)(35,43)(36,44)(37,45)(38,46)(39,47)(40,48)(49,57)(50,58)(51,59)(52,60)(53,61)(54,62)(55,63)(56,64), (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)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64), (1,26,59,38)(2,43,60,31)(3,40,61,28)(4,17,62,45)(5,30,63,42)(6,47,64,19)(7,44,49,32)(8,21,50,33)(9,18,51,46)(10,35,52,23)(11,48,53,20)(12,25,54,37)(13,22,55,34)(14,39,56,27)(15,36,57,24)(16,29,58,41) );`

`G=PermutationGroup([[(2,52),(4,54),(6,56),(8,58),(10,60),(12,62),(14,64),(16,50),(17,45),(18,26),(19,47),(20,28),(21,33),(22,30),(23,35),(24,32),(25,37),(27,39),(29,41),(31,43),(34,42),(36,44),(38,46),(40,48)], [(1,59),(2,60),(3,61),(4,62),(5,63),(6,64),(7,49),(8,50),(9,51),(10,52),(11,53),(12,54),(13,55),(14,56),(15,57),(16,58),(17,45),(18,46),(19,47),(20,48),(21,33),(22,34),(23,35),(24,36),(25,37),(26,38),(27,39),(28,40),(29,41),(30,42),(31,43),(32,44)], [(1,9),(2,10),(3,11),(4,12),(5,13),(6,14),(7,15),(8,16),(17,25),(18,26),(19,27),(20,28),(21,29),(22,30),(23,31),(24,32),(33,41),(34,42),(35,43),(36,44),(37,45),(38,46),(39,47),(40,48),(49,57),(50,58),(51,59),(52,60),(53,61),(54,62),(55,63),(56,64)], [(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),(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48),(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64)], [(1,26,59,38),(2,43,60,31),(3,40,61,28),(4,17,62,45),(5,30,63,42),(6,47,64,19),(7,44,49,32),(8,21,50,33),(9,18,51,46),(10,35,52,23),(11,48,53,20),(12,25,54,37),(13,22,55,34),(14,39,56,27),(15,36,57,24),(16,29,58,41)]])`

38 conjugacy classes

 class 1 2A 2B 2C 2D 4A 4B 4C 4D 4E 4F 4G 8A ··· 8H 8I 8J 16A ··· 16P order 1 2 2 2 2 4 4 4 4 4 4 4 8 ··· 8 8 8 16 ··· 16 size 1 1 1 1 4 1 1 1 1 4 8 8 2 ··· 2 8 8 4 ··· 4

38 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 4 4 type + + + + + - image C1 C2 C2 C4 C4 C8 D4 M4(2) D4.C8 C23⋊C4 C4.10D4 kernel C23.7M4(2) C22⋊C16 C42.6C22 C42⋊C2 C2×M4(2) C4⋊C4 C2×C8 C23 C2 C4 C4 # reps 1 2 1 2 2 8 2 2 16 1 1

Matrix representation of C23.7M4(2) in GL4(𝔽17) generated by

 1 0 0 0 13 16 0 0 0 0 1 0 0 0 2 16
,
 16 0 0 0 0 16 0 0 0 0 1 0 0 0 0 1
,
 16 0 0 0 0 16 0 0 0 0 16 0 0 0 0 16
,
 5 0 0 0 1 14 0 0 0 0 12 5 0 0 15 5
,
 13 15 0 0 0 4 0 0 0 0 16 1 0 0 0 1
`G:=sub<GL(4,GF(17))| [1,13,0,0,0,16,0,0,0,0,1,2,0,0,0,16],[16,0,0,0,0,16,0,0,0,0,1,0,0,0,0,1],[16,0,0,0,0,16,0,0,0,0,16,0,0,0,0,16],[5,1,0,0,0,14,0,0,0,0,12,15,0,0,5,5],[13,0,0,0,15,4,0,0,0,0,16,0,0,0,1,1] >;`

C23.7M4(2) in GAP, Magma, Sage, TeX

`C_2^3._7M_4(2)`
`% in TeX`

`G:=Group("C2^3.7M4(2)");`
`// GroupNames label`

`G:=SmallGroup(128,55);`
`// by ID`

`G=gap.SmallGroup(128,55);`
`# by ID`

`G:=PCGroup([7,-2,2,-2,2,-2,2,-2,56,85,120,422,723,352,1242,136,124]);`
`// Polycyclic`

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

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