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## G = C23.556C24order 128 = 27

### 273rd central stem extension by C23 of C24

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C23 — C23.556C24
 Chief series C1 — C2 — C22 — C23 — C24 — C22×D4 — C2×C4⋊D4 — C23.556C24
 Lower central C1 — C23 — C23.556C24
 Upper central C1 — C23 — C23.556C24
 Jennings C1 — C23 — C23.556C24

Generators and relations for C23.556C24
G = < a,b,c,d,e,f,g | a2=b2=c2=f2=g2=1, d2=b, e2=c, ab=ba, ac=ca, ede-1=ad=da, ae=ea, gfg=af=fa, ag=ga, bc=cb, fdf=bd=db, be=eb, bf=fb, bg=gb, cd=dc, fef=ce=ec, cf=fc, cg=gc, dg=gd, eg=ge >

Subgroups: 916 in 402 conjugacy classes, 116 normal (7 characteristic)
C1, C2, C2 [×6], C2 [×8], C4 [×16], C22, C22 [×10], C22 [×40], C2×C4 [×12], C2×C4 [×36], D4 [×40], C23, C23 [×6], C23 [×32], C22⋊C4 [×24], C4⋊C4 [×12], C22×C4 [×22], C22×C4 [×6], C2×D4 [×48], C24, C24 [×4], C2.C42 [×4], C2×C22⋊C4 [×12], C2×C4⋊C4 [×6], C4⋊D4 [×24], C23×C4 [×3], C22×D4 [×10], C2×C2.C42, C232D4 [×4], C23.4Q8 [×4], C2×C4⋊D4 [×6], C23.556C24
Quotients: C1, C2 [×15], C22 [×35], D4 [×12], C23 [×15], C2×D4 [×18], C24, C41D4 [×4], C22×D4 [×3], 2+ 1+4 [×3], 2- 1+4, C2×C41D4, C233D4 [×3], C22.31C24 [×3], C23.556C24

Smallest permutation representation of C23.556C24
On 64 points
Generators in S64
```(1 39)(2 40)(3 37)(4 38)(5 17)(6 18)(7 19)(8 20)(9 41)(10 42)(11 43)(12 44)(13 45)(14 46)(15 47)(16 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 62)(34 63)(35 64)(36 61)
(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 51)(2 52)(3 49)(4 50)(5 36)(6 33)(7 34)(8 35)(9 53)(10 54)(11 55)(12 56)(13 57)(14 58)(15 59)(16 60)(17 61)(18 62)(19 63)(20 64)(21 37)(22 38)(23 39)(24 40)(25 41)(26 42)(27 43)(28 44)(29 45)(30 46)(31 47)(32 48)
(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 59 51 15)(2 32 52 48)(3 57 49 13)(4 30 50 46)(5 41 36 25)(6 10 33 54)(7 43 34 27)(8 12 35 56)(9 61 53 17)(11 63 55 19)(14 38 58 22)(16 40 60 24)(18 42 62 26)(20 44 64 28)(21 45 37 29)(23 47 39 31)
(2 4)(5 61)(6 64)(7 63)(8 62)(9 41)(10 44)(11 43)(12 42)(13 57)(14 60)(15 59)(16 58)(17 36)(18 35)(19 34)(20 33)(22 24)(25 53)(26 56)(27 55)(28 54)(29 45)(30 48)(31 47)(32 46)(38 40)(50 52)
(1 11)(2 12)(3 9)(4 10)(5 45)(6 46)(7 47)(8 48)(13 17)(14 18)(15 19)(16 20)(21 25)(22 26)(23 27)(24 28)(29 36)(30 33)(31 34)(32 35)(37 41)(38 42)(39 43)(40 44)(49 53)(50 54)(51 55)(52 56)(57 61)(58 62)(59 63)(60 64)```

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

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

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

32 conjugacy classes

 class 1 2A ··· 2G 2H 2I 2J 2K 2L 2M 2N 2O 4A ··· 4L 4M 4N 4O 4P order 1 2 ··· 2 2 2 2 2 2 2 2 2 4 ··· 4 4 4 4 4 size 1 1 ··· 1 2 2 2 2 8 8 8 8 4 ··· 4 8 8 8 8

32 irreducible representations

 dim 1 1 1 1 1 2 4 4 type + + + + + + + - image C1 C2 C2 C2 C2 D4 2+ 1+4 2- 1+4 kernel C23.556C24 C2×C2.C42 C23⋊2D4 C23.4Q8 C2×C4⋊D4 C22×C4 C22 C22 # reps 1 1 4 4 6 12 3 1

Matrix representation of C23.556C24 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
,
 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 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 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
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 2 0 0 0 0 0 0 4 4 0 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 2 0
,
 4 3 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 4 3 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 4 0 0 0 0 0 0 0 0 4 0 0
,
 1 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 1 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
,
 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 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 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],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4],[4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,4,0,0,0,0,0,0,2,4,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,2,0],[4,1,0,0,0,0,0,0,3,1,0,0,0,0,0,0,0,0,4,1,0,0,0,0,0,0,3,1,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[1,4,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1],[4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0] >;`

C23.556C24 in GAP, Magma, Sage, TeX

`C_2^3._{556}C_2^4`
`% in TeX`

`G:=Group("C2^3.556C2^4");`
`// GroupNames label`

`G:=SmallGroup(128,1388);`
`// by ID`

`G=gap.SmallGroup(128,1388);`
`# by ID`

`G:=PCGroup([7,-2,2,2,2,-2,2,2,224,253,456,758,723,185]);`
`// Polycyclic`

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

׿
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