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## G = C23.199C24order 128 = 27

### 52nd 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 — C23.199C24
 Chief series C1 — C2 — C22 — C23 — C22×C4 — C2×C42 — C2×C42⋊C2 — C23.199C24
 Lower central C1 — C22 — C23.199C24
 Upper central C1 — C23 — C23.199C24
 Jennings C1 — C23 — C23.199C24

Generators and relations for C23.199C24
G = < a,b,c,d,e,f,g | a2=b2=c2=e2=1, d2=c, f2=b, g2=a, ab=ba, ac=ca, ede=gdg-1=ad=da, fef-1=ae=ea, af=fa, ag=ga, bc=cb, fdf-1=bd=db, be=eb, bf=fb, bg=gb, cd=dc, ce=ec, cf=fc, cg=gc, eg=ge, fg=gf >

Subgroups: 476 in 300 conjugacy classes, 180 normal (18 characteristic)
C1, C2 [×3], C2 [×4], C2 [×4], C4 [×8], C4 [×16], C22 [×3], C22 [×8], C22 [×12], C2×C4 [×36], C2×C4 [×40], C23, C23 [×6], C23 [×4], C42 [×8], C42 [×4], C22⋊C4 [×8], C22⋊C4 [×4], C4⋊C4 [×8], C4⋊C4 [×20], C22×C4 [×2], C22×C4 [×24], C22×C4 [×8], C24, C2.C42 [×8], C2×C42 [×4], C2×C22⋊C4 [×4], C2×C4⋊C4 [×2], C2×C4⋊C4 [×14], C2×C4⋊C4 [×4], C42⋊C2 [×8], C42⋊C2 [×4], C23×C4, C23×C4 [×2], C428C4 [×2], C429C4 [×2], C23.8Q8 [×4], C23.65C23 [×4], C22×C4⋊C4, C2×C42⋊C2 [×2], C23.199C24
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], D4 [×4], Q8 [×4], C23 [×15], C4⋊C4 [×16], C22×C4 [×14], C2×D4 [×6], C2×Q8 [×6], C24, C2×C4⋊C4 [×12], C23×C4, C22×D4, C22×Q8, 2+ 1+4 [×2], 2- 1+4 [×2], C22×C4⋊C4, C23.33C23 [×2], C22.29C24, C23.38C23, C23.41C23 [×2], C23.199C24

Smallest permutation representation of C23.199C24
On 64 points
Generators in S64
```(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 3)(2 12)(4 10)(5 7)(6 37)(8 39)(9 11)(13 15)(14 44)(16 42)(17 47)(18 20)(19 45)(21 23)(22 52)(24 50)(25 27)(26 56)(28 54)(29 59)(30 32)(31 57)(33 61)(34 36)(35 63)(38 40)(41 43)(46 48)(49 51)(53 55)(58 60)(62 64)
(1 59 51 45)(2 46 52 60)(3 57 49 47)(4 48 50 58)(5 54 62 44)(6 41 63 55)(7 56 64 42)(8 43 61 53)(9 31 23 17)(10 18 24 32)(11 29 21 19)(12 20 22 30)(13 33 27 39)(14 40 28 34)(15 35 25 37)(16 38 26 36)
(1 13 9 41)(2 42 10 14)(3 15 11 43)(4 44 12 16)(5 20 38 48)(6 45 39 17)(7 18 40 46)(8 47 37 19)(21 53 49 25)(22 26 50 54)(23 55 51 27)(24 28 52 56)(29 61 57 35)(30 36 58 62)(31 63 59 33)(32 34 60 64)```

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

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

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

44 conjugacy classes

 class 1 2A ··· 2G 2H 2I 2J 2K 4A ··· 4H 4I ··· 4AF order 1 2 ··· 2 2 2 2 2 4 ··· 4 4 ··· 4 size 1 1 ··· 1 2 2 2 2 2 ··· 2 4 ··· 4

44 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 4 4 type + + + + + + + + - + - image C1 C2 C2 C2 C2 C2 C2 C4 D4 Q8 2+ 1+4 2- 1+4 kernel C23.199C24 C42⋊8C4 C42⋊9C4 C23.8Q8 C23.65C23 C22×C4⋊C4 C2×C42⋊C2 C42⋊C2 C22×C4 C22×C4 C22 C22 # reps 1 2 2 4 4 1 2 16 4 4 2 2

Matrix representation of C23.199C24 in GL8(𝔽5)

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

`G:=sub<GL(8,GF(5))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4],[4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[0,2,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,4,4,0,0,0,0,0,1,0,0,0,0,0,0,4,1,0,1,0,0,0,0,0,4,0,4],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,2,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4],[2,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,1,4,0,0,0,0,0,1,0,0,0,0,0,0,1,4,0,4,0,0,0,0,0,4,0,4],[4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,2,0,2,0,0,0,0,4,1,4,1,0,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0] >;`

C23.199C24 in GAP, Magma, Sage, TeX

`C_2^3._{199}C_2^4`
`% in TeX`

`G:=Group("C2^3.199C2^4");`
`// GroupNames label`

`G:=SmallGroup(128,1049);`
`// by ID`

`G=gap.SmallGroup(128,1049);`
`# by ID`

`G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,758,219,184,675,80]);`
`// Polycyclic`

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

׿
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