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## G = C23.32M4(2)  order 128 = 27

### 9th non-split extension by C23 of M4(2) acting via M4(2)/C2×C4=C2

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C22 — C23.32M4(2)
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C2×C42 — C22×C42 — C23.32M4(2)
 Lower central C1 — C22 — C23.32M4(2)
 Upper central C1 — C22×C4 — C23.32M4(2)
 Jennings C1 — C2 — C2 — C22×C4 — C23.32M4(2)

Generators and relations for C23.32M4(2)
G = < a,b,c,d,e | a2=b2=c2=d8=1, e2=b, dad-1=ab=ba, ac=ca, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede-1=cd5 >

Subgroups: 308 in 190 conjugacy classes, 80 normal (16 characteristic)
C1, C2 [×3], C2 [×4], C2 [×4], C4 [×4], C4 [×10], C22 [×3], C22 [×8], C22 [×12], C8 [×4], C2×C4 [×16], C2×C4 [×34], C23, C23 [×6], C23 [×4], C42 [×8], C2×C8 [×12], C22×C4 [×2], C22×C4 [×12], C22×C4 [×14], C24, C22⋊C8 [×4], C2×C42 [×4], C2×C42 [×4], C22×C8 [×4], C23×C4 [×3], C22.7C42 [×4], C2×C22⋊C8 [×2], C22×C42, C23.32M4(2)
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C8 [×4], C2×C4 [×6], D4 [×4], C23, C22⋊C4 [×4], C2×C8 [×6], M4(2) [×6], C22×C4, C2×D4 [×2], C4○D4 [×4], C22⋊C8 [×4], C2×C22⋊C4, C42⋊C2 [×2], C22.D4 [×4], C22×C8, C2×M4(2) [×3], C23.34D4, C2×C22⋊C8, C24.4C4, C42.12C4 [×2], C42.6C4 [×2], C23.32M4(2)

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

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

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

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

56 conjugacy classes

 class 1 2A ··· 2G 2H 2I 2J 2K 4A ··· 4H 4I ··· 4AB 8A ··· 8P order 1 2 ··· 2 2 2 2 2 4 ··· 4 4 ··· 4 8 ··· 8 size 1 1 ··· 1 2 2 2 2 1 ··· 1 2 ··· 2 4 ··· 4

56 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 type + + + + + image C1 C2 C2 C2 C4 C4 C8 D4 M4(2) C4○D4 M4(2) kernel C23.32M4(2) C22.7C42 C2×C22⋊C8 C22×C42 C2×C42 C23×C4 C22×C4 C22×C4 C2×C4 C2×C4 C23 # reps 1 4 2 1 4 4 16 4 8 8 4

Matrix representation of C23.32M4(2) in GL5(𝔽17)

 16 0 0 0 0 0 1 0 0 0 0 9 16 0 0 0 0 0 16 0 0 0 0 0 16
,
 1 0 0 0 0 0 16 0 0 0 0 0 16 0 0 0 0 0 1 0 0 0 0 0 1
,
 16 0 0 0 0 0 16 0 0 0 0 0 16 0 0 0 0 0 1 0 0 0 0 0 1
,
 9 0 0 0 0 0 2 9 0 0 0 0 15 0 0 0 0 0 2 0 0 0 0 9 15
,
 16 0 0 0 0 0 13 0 0 0 0 0 13 0 0 0 0 0 1 9 0 0 0 0 16

`G:=sub<GL(5,GF(17))| [16,0,0,0,0,0,1,9,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,16],[1,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,1,0,0,0,0,0,1],[16,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,1,0,0,0,0,0,1],[9,0,0,0,0,0,2,0,0,0,0,9,15,0,0,0,0,0,2,9,0,0,0,0,15],[16,0,0,0,0,0,13,0,0,0,0,0,13,0,0,0,0,0,1,0,0,0,0,9,16] >;`

C23.32M4(2) in GAP, Magma, Sage, TeX

`C_2^3._{32}M_4(2)`
`% in TeX`

`G:=Group("C2^3.32M4(2)");`
`// GroupNames label`

`G:=SmallGroup(128,549);`
`// by ID`

`G=gap.SmallGroup(128,549);`
`# by ID`

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

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