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## G = C23.262C24order 128 = 27

### 115th central extension by C23 of C24

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C22 — C23.262C24
 Chief series C1 — C2 — C22 — C23 — C22×C4 — C2×C42 — C2×C4⋊1D4 — C23.262C24
 Lower central C1 — C22 — C23.262C24
 Upper central C1 — C23 — C23.262C24
 Jennings C1 — C23 — C23.262C24

Generators and relations for C23.262C24
G = < a,b,c,d,e,f,g | a2=b2=c2=d2=1, e2=d, f2=cb=bc, g2=b, eae-1=gag-1=ab=ba, ac=ca, ad=da, faf-1=abc, bd=db, be=eb, bf=fb, bg=gb, cd=dc, fef-1=ce=ec, cf=fc, cg=gc, de=ed, df=fd, dg=gd, geg-1=bce, fg=gf >

Subgroups: 732 in 324 conjugacy classes, 132 normal (6 characteristic)
C1, C2 [×7], C2 [×8], C4 [×14], C22, C22 [×6], C22 [×40], C2×C4 [×6], C2×C4 [×46], D4 [×24], C23, C23 [×8], C23 [×24], C42 [×4], C22⋊C4 [×12], C22×C4 [×11], C22×C4 [×12], C2×D4 [×24], C2×D4 [×12], C24 [×4], C2.C42 [×12], C2×C42, C2×C22⋊C4 [×12], C41D4 [×8], C23×C4 [×4], C22×D4 [×6], C425C4 [×2], C23.23D4 [×12], C2×C41D4, C23.262C24
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], C23 [×15], C22×C4 [×14], C24, C23×C4, 2+ 1+4 [×6], C22.11C24 [×3], C22.54C24 [×4], C23.262C24

Smallest permutation representation of C23.262C24
On 64 points
Generators in S64
```(1 6)(2 40)(3 8)(4 38)(5 12)(7 10)(9 39)(11 37)(13 17)(14 46)(15 19)(16 48)(18 42)(20 44)(21 35)(22 62)(23 33)(24 64)(25 29)(26 58)(27 31)(28 60)(30 54)(32 56)(34 52)(36 50)(41 45)(43 47)(49 61)(51 63)(53 57)(55 59)
(1 9)(2 10)(3 11)(4 12)(5 38)(6 39)(7 40)(8 37)(13 41)(14 42)(15 43)(16 44)(17 45)(18 46)(19 47)(20 48)(21 49)(22 50)(23 51)(24 52)(25 53)(26 54)(27 55)(28 56)(29 57)(30 58)(31 59)(32 60)(33 63)(34 64)(35 61)(36 62)
(1 51)(2 52)(3 49)(4 50)(5 62)(6 63)(7 64)(8 61)(9 23)(10 24)(11 21)(12 22)(13 27)(14 28)(15 25)(16 26)(17 31)(18 32)(19 29)(20 30)(33 39)(34 40)(35 37)(36 38)(41 55)(42 56)(43 53)(44 54)(45 59)(46 60)(47 57)(48 58)
(1 3)(2 4)(5 7)(6 8)(9 11)(10 12)(13 15)(14 16)(17 19)(18 20)(21 23)(22 24)(25 27)(26 28)(29 31)(30 32)(33 35)(34 36)(37 39)(38 40)(41 43)(42 44)(45 47)(46 48)(49 51)(50 52)(53 55)(54 56)(57 59)(58 60)(61 63)(62 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 31 23 45)(2 18 24 60)(3 29 21 47)(4 20 22 58)(5 54 36 16)(6 41 33 27)(7 56 34 14)(8 43 35 25)(9 59 51 17)(10 46 52 32)(11 57 49 19)(12 48 50 30)(13 63 55 39)(15 61 53 37)(26 62 44 38)(28 64 42 40)
(1 13 9 41)(2 56 10 28)(3 15 11 43)(4 54 12 26)(5 30 38 58)(6 45 39 17)(7 32 40 60)(8 47 37 19)(14 52 42 24)(16 50 44 22)(18 34 46 64)(20 36 48 62)(21 53 49 25)(23 55 51 27)(29 61 57 35)(31 63 59 33)```

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

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

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

38 conjugacy classes

 class 1 2A ··· 2G 2H ··· 2O 4A ··· 4V order 1 2 ··· 2 2 ··· 2 4 ··· 4 size 1 1 ··· 1 4 ··· 4 4 ··· 4

38 irreducible representations

 dim 1 1 1 1 1 4 type + + + + + image C1 C2 C2 C2 C4 2+ 1+4 kernel C23.262C24 C42⋊5C4 C23.23D4 C2×C4⋊1D4 C4⋊1D4 C22 # reps 1 2 12 1 16 6

Matrix representation of C23.262C24 in GL9(𝔽5)

 1 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 4 1 0 0 0 0 0 0 0 2 0 1 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 1 4 1 2 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 4
,
 1 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 4
,
 4 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1
,
 2 0 0 0 0 0 0 0 0 0 3 0 3 0 0 0 0 0 0 0 0 4 1 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 4 2 0 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 4 4
,
 1 0 0 0 0 0 0 0 0 0 1 3 0 0 0 0 0 0 0 1 4 0 0 0 0 0 0 0 0 2 0 1 0 0 0 0 0 3 2 4 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 4 1 2 0 0 0 0 0 4 0 0 0 0 0 0 0 0 1 4 0 1
,
 4 0 0 0 0 0 0 0 0 0 1 3 0 0 0 0 0 0 0 1 4 0 0 0 0 0 0 0 3 2 0 4 0 0 0 0 0 0 2 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 4 1 2 0 0 0 0 0 0 0 0 4

`G:=sub<GL(9,GF(5))| [1,0,0,0,0,0,0,0,0,0,4,4,2,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,1,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,2,0,0,4],[1,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,4],[4,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1],[2,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,3,4,2,2,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,1,0,1,0,0,0,0,0,0,0,1,4,0,0,0,0,0,0,0,0,4],[1,0,0,0,0,0,0,0,0,0,1,1,0,3,0,0,0,0,0,3,4,2,2,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,4,1,0,0,0,0,0,0,4,0,4,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,2,0,1],[4,0,0,0,0,0,0,0,0,0,1,1,3,0,0,0,0,0,0,3,4,2,2,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,1,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,2,4] >;`

C23.262C24 in GAP, Magma, Sage, TeX

`C_2^3._{262}C_2^4`
`% in TeX`

`G:=Group("C2^3.262C2^4");`
`// GroupNames label`

`G:=SmallGroup(128,1112);`
`// by ID`

`G=gap.SmallGroup(128,1112);`
`# by ID`

`G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,758,555,268,1571,346,80]);`
`// Polycyclic`

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

׿
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