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## G = C24.219C23order 128 = 27

### 59th non-split extension by C24 of C23 acting via C23/C2=C22

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C22 — C24.219C23
 Chief series C1 — C2 — C22 — C23 — C22×C4 — C2×C42 — C4×C4⋊C4 — C24.219C23
 Lower central C1 — C22 — C24.219C23
 Upper central C1 — C23 — C24.219C23
 Jennings C1 — C23 — C24.219C23

Generators and relations for C24.219C23
G = < a,b,c,d,e,f,g | a2=b2=c2=1, d2=c, e2=ca=ac, f2=b, g2=a, ab=ba, ede-1=ad=da, geg-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, dg=gd, ef=fe, fg=gf >

Subgroups: 796 in 416 conjugacy classes, 164 normal (12 characteristic)
C1, C2 [×3], C2 [×4], C2 [×8], C4 [×8], C4 [×14], C22 [×3], C22 [×4], C22 [×40], C2×C4 [×22], C2×C4 [×38], D4 [×40], C23, C23 [×8], C23 [×24], C42 [×4], C42 [×8], C22⋊C4 [×20], C4⋊C4 [×8], C4⋊C4 [×4], C22×C4, C22×C4 [×10], C22×C4 [×20], C2×D4 [×24], C2×D4 [×20], C24 [×4], C2.C42 [×4], C2×C42, C2×C42 [×4], C2×C22⋊C4 [×12], C2×C4⋊C4 [×4], C4×D4 [×16], C41D4 [×8], C23×C4 [×4], C22×D4 [×6], C4×C4⋊C4 [×2], C23.23D4 [×4], C24.3C22 [×4], C2×C4×D4 [×4], C2×C41D4, C24.219C23
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], D4 [×8], C23 [×15], C22×C4 [×14], C2×D4 [×12], C4○D4 [×4], C24, C4×D4 [×8], C23×C4, C22×D4 [×2], C2×C4○D4 [×2], 2+ 1+4 [×2], C2×C4×D4 [×2], C22.11C24, D42, Q86D4 [×2], C22.53C24, C24.219C23

Smallest permutation representation of C24.219C23
On 64 points
Generators in S64
```(1 26)(2 27)(3 28)(4 25)(5 54)(6 55)(7 56)(8 53)(9 46)(10 47)(11 48)(12 45)(13 50)(14 51)(15 52)(16 49)(17 39)(18 40)(19 37)(20 38)(21 43)(22 44)(23 41)(24 42)(29 35)(30 36)(31 33)(32 34)(57 64)(58 61)(59 62)(60 63)
(1 61)(2 62)(3 63)(4 64)(5 33)(6 34)(7 35)(8 36)(9 41)(10 42)(11 43)(12 44)(13 37)(14 38)(15 39)(16 40)(17 52)(18 49)(19 50)(20 51)(21 48)(22 45)(23 46)(24 47)(25 57)(26 58)(27 59)(28 60)(29 56)(30 53)(31 54)(32 55)
(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 23)(2 22 25 42)(3 41 26 21)(4 24 27 44)(5 16 56 51)(6 50 53 15)(7 14 54 49)(8 52 55 13)(9 58 48 63)(10 62 45 57)(11 60 46 61)(12 64 47 59)(17 32 37 36)(18 35 38 31)(19 30 39 34)(20 33 40 29)
(1 49 61 18)(2 19 62 50)(3 51 63 20)(4 17 64 52)(5 9 33 41)(6 42 34 10)(7 11 35 43)(8 44 36 12)(13 27 37 59)(14 60 38 28)(15 25 39 57)(16 58 40 26)(21 56 48 29)(22 30 45 53)(23 54 46 31)(24 32 47 55)
(1 36 26 30)(2 33 27 31)(3 34 28 32)(4 35 25 29)(5 59 54 62)(6 60 55 63)(7 57 56 64)(8 58 53 61)(9 13 46 50)(10 14 47 51)(11 15 48 52)(12 16 45 49)(17 43 39 21)(18 44 40 22)(19 41 37 23)(20 42 38 24)```

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

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

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

50 conjugacy classes

 class 1 2A ··· 2G 2H ··· 2O 4A ··· 4X 4Y ··· 4AH 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 type + + + + + + + + image C1 C2 C2 C2 C2 C2 C4 D4 C4○D4 2+ 1+4 kernel C24.219C23 C4×C4⋊C4 C23.23D4 C24.3C22 C2×C4×D4 C2×C4⋊1D4 C4⋊1D4 C4⋊C4 C2×C4 C22 # reps 1 2 4 4 4 1 16 8 8 2

Matrix representation of C24.219C23 in GL5(𝔽5)

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

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

C24.219C23 in GAP, Magma, Sage, TeX

`C_2^4._{219}C_2^3`
`% in TeX`

`G:=Group("C2^4.219C2^3");`
`// GroupNames label`

`G:=SmallGroup(128,1098);`
`// by ID`

`G=gap.SmallGroup(128,1098);`
`# by ID`

`G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,568,758,184,346,80]);`
`// Polycyclic`

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

׿
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