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## G = C23.9M4(2)  order 128 = 27

### 5th non-split extension by C23 of M4(2) acting via M4(2)/C4=C22

p-group, metabelian, nilpotent (class 2), monomial

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C23 — C23.9M4(2)
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C2×C42 — C2×C8⋊C4 — C23.9M4(2)
 Lower central C1 — C23 — C23.9M4(2)
 Upper central C1 — C22×C4 — C23.9M4(2)
 Jennings C1 — C2 — C2 — C22×C4 — C23.9M4(2)

Generators and relations for C23.9M4(2)
G = < a,b,c,d,e | a2=b2=c2=d8=1, e2=b, ab=ba, eae-1=ac=ca, dad-1=abc, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede-1=cd5 >

Subgroups: 252 in 140 conjugacy classes, 60 normal (52 characteristic)
C1, C2 [×7], C2 [×2], C4 [×4], C4 [×6], C22 [×7], C22 [×10], C8 [×6], C2×C4 [×6], C2×C4 [×2], C2×C4 [×18], C23, C23 [×2], C23 [×6], C42 [×4], C22⋊C4 [×4], C2×C8 [×4], C2×C8 [×10], C22×C4 [×6], C22×C4 [×6], C24, C2.C42 [×2], C8⋊C4 [×2], C22⋊C8 [×4], C4⋊C8 [×2], C2×C42 [×2], C2×C22⋊C4 [×2], C22×C8 [×4], C23×C4, C22.7C42 [×2], C4×C22⋊C4, C2×C8⋊C4, C2×C22⋊C8 [×2], C2×C4⋊C8, C23.9M4(2)
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, M4(2) [×4], C22×C4, C2×D4 [×2], C4○D4 [×4], C42⋊C2, C4×D4 [×2], C4⋊D4, C22.D4, C4.4D4, C422C2, C2×M4(2) [×2], C8○D4 [×2], C24.C22, C42.6C4, C42.7C22, C89D4 [×3], C86D4, C23.9M4(2)

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

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

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

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

44 conjugacy classes

 class 1 2A ··· 2G 2H 2I 4A ··· 4H 4I ··· 4R 8A ··· 8P order 1 2 ··· 2 2 2 4 ··· 4 4 ··· 4 8 ··· 8 size 1 1 ··· 1 4 4 1 ··· 1 4 ··· 4 4 ··· 4

44 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 type + + + + + + + image C1 C2 C2 C2 C2 C2 C4 C4 D4 M4(2) C4○D4 M4(2) C8○D4 kernel C23.9M4(2) C22.7C42 C4×C22⋊C4 C2×C8⋊C4 C2×C22⋊C8 C2×C4⋊C8 C2.C42 C2×C22⋊C4 C2×C8 C2×C4 C2×C4 C23 C22 # reps 1 2 1 1 2 1 4 4 4 4 8 4 8

Matrix representation of C23.9M4(2) in GL6(𝔽17)

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

`G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,16,0,0,0,0,0,0,1,13,0,0,0,0,0,16,0,0,0,0,0,0,1,15,0,0,0,0,0,16],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[0,16,0,0,0,0,4,0,0,0,0,0,0,0,1,0,0,0,0,0,9,16,0,0,0,0,0,0,9,16,0,0,0,0,9,8],[4,0,0,0,0,0,0,13,0,0,0,0,0,0,13,0,0,0,0,0,0,13,0,0,0,0,0,0,13,8,0,0,0,0,13,4] >;`

C23.9M4(2) in GAP, Magma, Sage, TeX

`C_2^3._9M_4(2)`
`% in TeX`

`G:=Group("C2^3.9M4(2)");`
`// GroupNames label`

`G:=SmallGroup(128,656);`
`// by ID`

`G=gap.SmallGroup(128,656);`
`# by ID`

`G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,288,422,723,58,124]);`
`// Polycyclic`

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

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