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## G = C23.223C24order 128 = 27

### 76th 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 — C23.223C24
 Chief series C1 — C2 — C22 — C23 — C22×C4 — C23×C4 — C22×C4○D4 — C23.223C24
 Lower central C1 — C22 — C23.223C24
 Upper central C1 — C23 — C23.223C24
 Jennings C1 — C23 — C23.223C24

Generators and relations for C23.223C24
G = < a,b,c,d,e,f,g | a2=b2=c2=d2=1, e2=d, f2=g2=b, gag-1=ab=ba, ac=ca, ad=da, eae-1=abc, af=fa, bc=cb, bd=db, fef-1=be=eb, gfg-1=bf=fb, bg=gb, cd=dc, ce=ec, cf=fc, cg=gc, de=ed, df=fd, dg=gd, eg=ge >

Subgroups: 748 in 436 conjugacy classes, 184 normal (13 characteristic)
C1, C2 [×3], C2 [×4], C2 [×6], C4 [×12], C4 [×12], C22 [×3], C22 [×4], C22 [×30], C2×C4 [×24], C2×C4 [×48], D4 [×24], Q8 [×16], C23, C23 [×6], C23 [×18], C42 [×12], C22⋊C4 [×18], C4⋊C4 [×12], C22×C4, C22×C4 [×23], C22×C4 [×12], C2×D4 [×12], C2×D4 [×12], C2×Q8 [×12], C4○D4 [×32], C24 [×3], C2.C42, C2.C42 [×3], C2×C42 [×6], C2×C22⋊C4 [×12], C2×C4⋊C4 [×6], C4×Q8 [×8], C23×C4 [×3], C22×D4 [×3], C22×Q8, C2×C4○D4 [×8], C2×C4○D4 [×8], C4×C22⋊C4 [×3], C23.7Q8 [×3], C24.3C22 [×6], C2×C4×Q8 [×2], C22×C4○D4, C23.223C24
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], D4 [×8], C23 [×15], C22⋊C4 [×16], C22×C4 [×14], C2×D4 [×12], C4○D4 [×4], C24, C2×C22⋊C4 [×12], C23×C4, C22×D4 [×2], C2×C4○D4 [×2], 2+ 1+4, 2- 1+4, C22×C22⋊C4, C4×C4○D4, C23.33C23, Q85D4 [×2], Q86D4 [×2], C23.223C24

Smallest permutation representation of C23.223C24
On 64 points
Generators in S64
```(1 43)(2 28)(3 41)(4 26)(5 58)(6 19)(7 60)(8 17)(9 13)(10 54)(11 15)(12 56)(14 50)(16 52)(18 62)(20 64)(21 25)(22 42)(23 27)(24 44)(29 35)(30 38)(31 33)(32 40)(34 48)(36 46)(37 45)(39 47)(49 53)(51 55)(57 61)(59 63)
(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 39 11 6)(2 7 12 40)(3 37 9 8)(4 5 10 38)(13 17 41 45)(14 46 42 18)(15 19 43 47)(16 48 44 20)(21 35 49 61)(22 62 50 36)(23 33 51 63)(24 64 52 34)(25 29 53 57)(26 58 54 30)(27 31 55 59)(28 60 56 32)
(1 15 11 43)(2 16 12 44)(3 13 9 41)(4 14 10 42)(5 18 38 46)(6 19 39 47)(7 20 40 48)(8 17 37 45)(21 53 49 25)(22 54 50 26)(23 55 51 27)(24 56 52 28)(29 61 57 35)(30 62 58 36)(31 63 59 33)(32 64 60 34)```

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

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

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

50 conjugacy classes

 class 1 2A ··· 2G 2H ··· 2M 4A ··· 4X 4Y ··· 4AJ order 1 2 ··· 2 2 ··· 2 4 ··· 4 4 ··· 4 size 1 1 ··· 1 4 ··· 4 2 ··· 2 4 ··· 4

50 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 4 4 type + + + + + + + + - image C1 C2 C2 C2 C2 C2 C4 D4 C4○D4 2+ 1+4 2- 1+4 kernel C23.223C24 C4×C22⋊C4 C23.7Q8 C24.3C22 C2×C4×Q8 C22×C4○D4 C2×C4○D4 C2×Q8 C2×C4 C22 C22 # reps 1 3 3 6 2 1 16 8 8 1 1

Matrix representation of C23.223C24 in GL5(𝔽5)

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

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

C23.223C24 in GAP, Magma, Sage, TeX

`C_2^3._{223}C_2^4`
`% in TeX`

`G:=Group("C2^3.223C2^4");`
`// GroupNames label`

`G:=SmallGroup(128,1073);`
`// by ID`

`G=gap.SmallGroup(128,1073);`
`# by ID`

`G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,120,758,100,346,80]);`
`// Polycyclic`

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

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