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G = C23.179C24order 128 = 27

32nd 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 — C23.179C24
 Chief series C1 — C2 — C22 — C23 — C22×C4 — C23×C4 — C22×C42 — C23.179C24
 Lower central C1 — C22 — C23.179C24
 Upper central C1 — C22×C4 — C23.179C24
 Jennings C1 — C23 — C23.179C24

Generators and relations for C23.179C24
G = < a,b,c,d,e,f,g | a2=b2=c2=d2=1, e2=d, f2=db=bd, g2=c, ab=ba, eae-1=ac=ca, ad=da, af=fa, ag=ga, bc=cb, fef-1=be=eb, bf=fb, bg=gb, cd=dc, ce=ec, cf=fc, cg=gc, de=ed, df=fd, dg=gd, eg=ge, fg=gf >

Subgroups: 780 in 468 conjugacy classes, 188 normal (16 characteristic)
C1, C2, C2 [×6], C2 [×8], C4 [×8], C4 [×16], C22 [×3], C22 [×8], C22 [×32], C2×C4 [×40], C2×C4 [×56], D4 [×24], Q8 [×8], C23, C23 [×10], C23 [×16], C42 [×12], C22⋊C4 [×20], C4⋊C4 [×4], C22×C4 [×2], C22×C4 [×30], C22×C4 [×20], C2×D4 [×12], C2×D4 [×12], C2×Q8 [×4], C2×Q8 [×4], C4○D4 [×32], C24, C24 [×2], C2.C42 [×8], C2×C42 [×6], C2×C42 [×4], C2×C22⋊C4 [×10], C2×C4⋊C4 [×2], C42⋊C2 [×4], C23×C4, C23×C4 [×4], C22×D4, C22×D4 [×2], C22×Q8, C2×C4○D4 [×8], C2×C4○D4 [×8], C4×C22⋊C4 [×4], C23.23D4 [×4], C24.3C22 [×2], C23.67C23 [×2], C22×C42, C2×C42⋊C2, C22×C4○D4, C23.179C24
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 [×8], C24, C2×C22⋊C4 [×12], C23×C4, C22×D4 [×2], C2×C4○D4 [×4], C22×C22⋊C4, C4×C4○D4 [×2], C22.19C24 [×2], C22.26C24 [×2], C23.179C24

Smallest permutation representation of C23.179C24
On 64 points
Generators in S64
```(1 49)(2 39)(3 51)(4 37)(5 11)(6 42)(7 9)(8 44)(10 29)(12 31)(13 57)(14 26)(15 59)(16 28)(17 60)(18 25)(19 58)(20 27)(21 35)(22 55)(23 33)(24 53)(30 41)(32 43)(34 45)(36 47)(38 61)(40 63)(46 54)(48 56)(50 62)(52 64)
(1 26)(2 27)(3 28)(4 25)(5 54)(6 55)(7 56)(8 53)(9 48)(10 45)(11 46)(12 47)(13 52)(14 49)(15 50)(16 51)(17 40)(18 37)(19 38)(20 39)(21 41)(22 42)(23 43)(24 44)(29 34)(30 35)(31 36)(32 33)(57 64)(58 61)(59 62)(60 63)
(1 61)(2 62)(3 63)(4 64)(5 30)(6 31)(7 32)(8 29)(9 43)(10 44)(11 41)(12 42)(13 18)(14 19)(15 20)(16 17)(21 46)(22 47)(23 48)(24 45)(25 57)(26 58)(27 59)(28 60)(33 56)(34 53)(35 54)(36 55)(37 52)(38 49)(39 50)(40 51)
(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 43 28 21)(2 24 25 42)(3 41 26 23)(4 22 27 44)(5 19 56 40)(6 39 53 18)(7 17 54 38)(8 37 55 20)(9 60 46 61)(10 64 47 59)(11 58 48 63)(12 62 45 57)(13 31 50 34)(14 33 51 30)(15 29 52 36)(16 35 49 32)
(1 34 61 53)(2 35 62 54)(3 36 63 55)(4 33 64 56)(5 27 30 59)(6 28 31 60)(7 25 32 57)(8 26 29 58)(9 18 43 13)(10 19 44 14)(11 20 41 15)(12 17 42 16)(21 50 46 39)(22 51 47 40)(23 52 48 37)(24 49 45 38)```

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

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

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

56 conjugacy classes

 class 1 2A ··· 2G 2H 2I 2J 2K 2L 2M 2N 2O 4A ··· 4H 4I ··· 4AB 4AC ··· 4AN 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 4 4 4 4 1 ··· 1 2 ··· 2 4 ··· 4

56 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 type + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C4 D4 C4○D4 kernel C23.179C24 C4×C22⋊C4 C23.23D4 C24.3C22 C23.67C23 C22×C42 C2×C42⋊C2 C22×C4○D4 C2×C4○D4 C22×C4 C2×C4 # reps 1 4 4 2 2 1 1 1 16 8 16

Matrix representation of C23.179C24 in GL5(𝔽5)

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

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

C23.179C24 in GAP, Magma, Sage, TeX

`C_2^3._{179}C_2^4`
`% in TeX`

`G:=Group("C2^3.179C2^4");`
`// GroupNames label`

`G:=SmallGroup(128,1029);`
`// by ID`

`G=gap.SmallGroup(128,1029);`
`# by ID`

`G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,568,758,136]);`
`// Polycyclic`

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

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