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## G = C23.259C24order 128 = 27

### 112nd 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 — C23.259C24
 Chief series C1 — C2 — C22 — C23 — C22×C4 — C23×C4 — C2×C4⋊D4 — C23.259C24
 Lower central C1 — C22 — C23.259C24
 Upper central C1 — C23 — C23.259C24
 Jennings C1 — C23 — C23.259C24

Generators and relations for C23.259C24
G = < a,b,c,d,e,f,g | a2=b2=c2=d2=f2=1, e2=d, g2=b, eae-1=gag-1=ab=ba, ac=ca, ad=da, faf=abc, bc=cb, bd=db, be=eb, bf=fb, bg=gb, cd=dc, fef=ce=ec, cf=fc, cg=gc, de=ed, df=fd, dg=gd, geg-1=bce, fg=gf >

Subgroups: 604 in 288 conjugacy classes, 132 normal (22 characteristic)
C1, C2 [×3], C2 [×4], C2 [×6], C4 [×16], C22 [×3], C22 [×4], C22 [×30], C2×C4 [×8], C2×C4 [×44], D4 [×12], C23, C23 [×6], C23 [×18], C42 [×2], C22⋊C4 [×8], C22⋊C4 [×10], C4⋊C4 [×4], C4⋊C4 [×4], C22×C4 [×2], C22×C4 [×14], C22×C4 [×8], C2×D4 [×12], C2×D4 [×6], C24, C24 [×2], C2.C42 [×2], C2.C42 [×8], C2×C42 [×2], C2×C22⋊C4 [×2], C2×C22⋊C4 [×10], C2×C4⋊C4, C2×C4⋊C4 [×4], C4⋊D4 [×8], C23×C4, C23×C4 [×2], C22×D4, C22×D4 [×2], C23.7Q8, C23.34D4, C23.8Q8 [×2], C23.23D4 [×4], C23.63C23 [×2], C24.C22 [×2], C24.3C22 [×2], C2×C4⋊D4, C23.259C24
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], C23 [×15], C22×C4 [×14], C24, C23×C4, 2+ 1+4 [×5], 2- 1+4, C22.11C24 [×2], C23.33C23, C22.54C24 [×2], C22.56C24 [×2], C23.259C24

Smallest permutation representation of C23.259C24
On 64 points
Generators in S64
```(1 43)(2 16)(3 41)(4 14)(5 30)(6 59)(7 32)(8 57)(9 13)(10 42)(11 15)(12 44)(17 61)(18 36)(19 63)(20 34)(21 25)(22 54)(23 27)(24 56)(26 50)(28 52)(29 37)(31 39)(33 47)(35 45)(38 58)(40 60)(46 62)(48 64)(49 53)(51 55)
(1 11)(2 12)(3 9)(4 10)(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 36)(6 33)(7 34)(8 35)(9 21)(10 22)(11 23)(12 24)(13 25)(14 26)(15 27)(16 28)(17 29)(18 30)(19 31)(20 32)(37 61)(38 62)(39 63)(40 64)(41 53)(42 54)(43 55)(44 56)(45 57)(46 58)(47 59)(48 60)
(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 47)(2 60)(3 45)(4 58)(5 26)(6 15)(7 28)(8 13)(9 17)(10 30)(11 19)(12 32)(14 36)(16 34)(18 22)(20 24)(21 29)(23 31)(25 35)(27 33)(37 41)(38 54)(39 43)(40 56)(42 62)(44 64)(46 50)(48 52)(49 57)(51 59)(53 61)(55 63)
(1 15 11 43)(2 56 12 28)(3 13 9 41)(4 54 10 26)(5 58 38 30)(6 19 39 47)(7 60 40 32)(8 17 37 45)(14 50 42 22)(16 52 44 24)(18 36 46 62)(20 34 48 64)(21 53 49 25)(23 55 51 27)(29 61 57 35)(31 63 59 33)```

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

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

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

38 conjugacy classes

 class 1 2A ··· 2G 2H ··· 2M 4A ··· 4X 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 1 1 1 1 1 4 4 type + + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C2 C2 C2 C4 2+ 1+4 2- 1+4 kernel C23.259C24 C23.7Q8 C23.34D4 C23.8Q8 C23.23D4 C23.63C23 C24.C22 C24.3C22 C2×C4⋊D4 C4⋊D4 C22 C22 # reps 1 1 1 2 4 2 2 2 1 16 5 1

Matrix representation of C23.259C24 in GL9(𝔽5)

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

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

C23.259C24 in GAP, Magma, Sage, TeX

`C_2^3._{259}C_2^4`
`% in TeX`

`G:=Group("C2^3.259C2^4");`
`// GroupNames label`

`G:=SmallGroup(128,1109);`
`// by ID`

`G=gap.SmallGroup(128,1109);`
`# by ID`

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

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