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## G = C24.268C23order 128 = 27

### 108th 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 — C23 — C24.268C23
 Chief series C1 — C2 — C22 — C23 — C22×C4 — C2×C42 — C4×C22⋊C4 — C24.268C23
 Lower central C1 — C23 — C24.268C23
 Upper central C1 — C23 — C24.268C23
 Jennings C1 — C23 — C24.268C23

Generators and relations for C24.268C23
G = < a,b,c,d,e,f,g | a2=b2=c2=d2=1, e2=c, f2=cb=bc, g2=b, eae-1=ab=ba, ac=ca, faf-1=ad=da, ag=ga, bd=db, geg-1=be=eb, bf=fb, bg=gb, cd=dc, fef-1=ce=ec, cf=fc, cg=gc, de=ed, df=fd, dg=gd, fg=gf >

Subgroups: 420 in 246 conjugacy classes, 112 normal (42 characteristic)
C1, C2 [×7], C2 [×2], C4 [×4], C4 [×18], C22 [×7], C22 [×10], C2×C4 [×16], C2×C4 [×42], C23, C23 [×2], C23 [×6], C42 [×8], C22⋊C4 [×10], C4⋊C4 [×4], C4⋊C4 [×26], C22×C4 [×6], C22×C4 [×12], C22×C4 [×4], C24, C2.C42 [×2], C2.C42 [×6], C2×C42 [×2], C2×C42 [×2], C2×C22⋊C4 [×2], C2×C22⋊C4 [×4], C2×C4⋊C4 [×4], C2×C4⋊C4 [×12], C42⋊C2 [×4], C42.C2 [×4], C23×C4, C4×C22⋊C4, C23.7Q8, C429C4, C23.63C23 [×2], C23.65C23 [×2], C23.65C23 [×2], C23.Q8 [×2], C23.4Q8 [×2], C2×C42⋊C2, C2×C42.C2, C24.268C23
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], Q8 [×4], C23 [×15], C2×D4 [×6], C2×Q8 [×6], C4○D4 [×6], C24, C22⋊Q8 [×4], C22×D4, C22×Q8, C2×C4○D4 [×3], 2+ 1+4, 2- 1+4, C2×C22⋊Q8, C23.37C23, C23.41C23, D46D4, Q86D4, C22.46C24, C22.47C24, C24.268C23

Smallest permutation representation of C24.268C23
On 64 points
Generators in S64
```(2 40)(4 38)(5 36)(6 62)(7 34)(8 64)(10 42)(12 44)(13 57)(14 30)(15 59)(16 32)(17 61)(18 33)(19 63)(20 35)(22 50)(24 52)(26 54)(28 56)(29 45)(31 47)(46 58)(48 60)
(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 45 37 15)(2 48 38 14)(3 47 39 13)(4 46 40 16)(5 41 19 11)(6 44 20 10)(7 43 17 9)(8 42 18 12)(21 59 51 29)(22 58 52 32)(23 57 49 31)(24 60 50 30)(25 63 55 36)(26 62 56 35)(27 61 53 34)(28 64 54 33)
(1 43 39 11)(2 12 40 44)(3 41 37 9)(4 10 38 42)(5 45 17 13)(6 14 18 46)(7 47 19 15)(8 16 20 48)(21 53 49 25)(22 26 50 54)(23 55 51 27)(24 28 52 56)(29 61 57 36)(30 33 58 62)(31 63 59 34)(32 35 60 64)```

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

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

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

38 conjugacy classes

 class 1 2A ··· 2G 2H 2I 4A ··· 4H 4I ··· 4X 4Y 4Z 4AA 4AB order 1 2 ··· 2 2 2 4 ··· 4 4 ··· 4 4 4 4 4 size 1 1 ··· 1 4 4 2 ··· 2 4 ··· 4 8 8 8 8

38 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 4 4 type + + + + + + + + + + + - + - image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 D4 Q8 C4○D4 2+ 1+4 2- 1+4 kernel C24.268C23 C4×C22⋊C4 C23.7Q8 C42⋊9C4 C23.63C23 C23.65C23 C23.Q8 C23.4Q8 C2×C42⋊C2 C2×C42.C2 C4⋊C4 C22×C4 C2×C4 C22 C22 # reps 1 1 1 1 2 4 2 2 1 1 4 4 12 1 1

Matrix representation of C24.268C23 in GL6(𝔽5)

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

`G:=sub<GL(6,GF(5))| [1,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,4],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[2,0,0,0,0,0,0,3,0,0,0,0,0,0,2,0,0,0,0,0,2,3,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[0,4,0,0,0,0,1,0,0,0,0,0,0,0,4,2,0,0,0,0,4,1,0,0,0,0,0,0,2,0,0,0,0,0,0,2],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,2,0,0,0,0,0,0,3] >;`

C24.268C23 in GAP, Magma, Sage, TeX

`C_2^4._{268}C_2^3`
`% in TeX`

`G:=Group("C2^4.268C2^3");`
`// GroupNames label`

`G:=SmallGroup(128,1173);`
`// by ID`

`G=gap.SmallGroup(128,1173);`
`# by ID`

`G:=PCGroup([7,-2,2,2,2,-2,2,2,224,253,758,723,268,675,80]);`
`// Polycyclic`

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

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