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## G = C24.345C23order 128 = 27

### 185th 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 — C24.345C23
 Chief series C1 — C2 — C22 — C23 — C22×C4 — C2×C42 — C4×C4⋊C4 — C24.345C23
 Lower central C1 — C23 — C24.345C23
 Upper central C1 — C23 — C24.345C23
 Jennings C1 — C23 — C24.345C23

Generators and relations for C24.345C23
G = < a,b,c,d,e,f,g | a2=b2=c2=d2=1, e2=bcd, f2=d, g2=c, eae-1=gag-1=ab=ba, faf-1=ac=ca, ad=da, bc=cb, bd=db, be=eb, bf=fb, bg=gb, cd=dc, ce=ec, cf=fc, cg=gc, fef-1=de=ed, df=fd, dg=gd, eg=ge, fg=gf >

Subgroups: 388 in 214 conjugacy classes, 100 normal (82 characteristic)
C1, C2 [×7], C2 [×2], C4 [×20], C22 [×7], C22 [×10], C2×C4 [×12], C2×C4 [×40], C23, C23 [×2], C23 [×6], C42 [×5], C22⋊C4 [×4], C22⋊C4 [×8], C4⋊C4 [×19], C22×C4 [×14], C22×C4 [×4], C24, C2.C42 [×12], C2×C42 [×4], C2×C22⋊C4 [×6], C2×C4⋊C4 [×12], C23×C4, C4×C22⋊C4, C4×C4⋊C4, C23.8Q8 [×2], C23.63C23 [×2], C24.C22 [×2], C23.65C23 [×3], C23.Q8 [×2], C23.81C23, C23.83C23, C24.345C23
Quotients: C1, C2 [×15], C22 [×35], Q8 [×4], C23 [×15], C2×Q8 [×6], C4○D4 [×8], C24, C22×Q8, C2×C4○D4 [×4], 2+ 1+4 [×2], C23.36C23, C23.37C23, C22.32C24, C22.47C24 [×2], D43Q8 [×2], C24.345C23

Smallest permutation representation of C24.345C23
On 64 points
Generators in S64
```(2 56)(4 54)(5 23)(6 25)(7 21)(8 27)(9 43)(11 41)(13 45)(15 47)(17 62)(18 36)(19 64)(20 34)(22 40)(24 38)(26 39)(28 37)(29 58)(31 60)(33 51)(35 49)(50 63)(52 61)
(1 55)(2 56)(3 53)(4 54)(5 37)(6 38)(7 39)(8 40)(9 43)(10 44)(11 41)(12 42)(13 45)(14 46)(15 47)(16 48)(17 49)(18 50)(19 51)(20 52)(21 26)(22 27)(23 28)(24 25)(29 58)(30 59)(31 60)(32 57)(33 64)(34 61)(35 62)(36 63)
(1 12)(2 9)(3 10)(4 11)(5 28)(6 25)(7 26)(8 27)(13 31)(14 32)(15 29)(16 30)(17 35)(18 36)(19 33)(20 34)(21 39)(22 40)(23 37)(24 38)(41 54)(42 55)(43 56)(44 53)(45 60)(46 57)(47 58)(48 59)(49 62)(50 63)(51 64)(52 61)
(1 44)(2 41)(3 42)(4 43)(5 21)(6 22)(7 23)(8 24)(9 54)(10 55)(11 56)(12 53)(13 58)(14 59)(15 60)(16 57)(17 64)(18 61)(19 62)(20 63)(25 40)(26 37)(27 38)(28 39)(29 45)(30 46)(31 47)(32 48)(33 49)(34 50)(35 51)(36 52)
(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 52 44 36)(2 33 41 49)(3 50 42 34)(4 35 43 51)(5 58 21 13)(6 14 22 59)(7 60 23 15)(8 16 24 57)(9 19 54 62)(10 63 55 20)(11 17 56 64)(12 61 53 18)(25 32 40 48)(26 45 37 29)(27 30 38 46)(28 47 39 31)
(1 13 12 31)(2 14 9 32)(3 15 10 29)(4 16 11 30)(5 61 28 52)(6 62 25 49)(7 63 26 50)(8 64 27 51)(17 38 35 24)(18 39 36 21)(19 40 33 22)(20 37 34 23)(41 59 54 48)(42 60 55 45)(43 57 56 46)(44 58 53 47)```

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

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

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

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 4 type + + + + + + + + + + - + image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 Q8 C4○D4 2+ 1+4 kernel C24.345C23 C4×C22⋊C4 C4×C4⋊C4 C23.8Q8 C23.63C23 C24.C22 C23.65C23 C23.Q8 C23.81C23 C23.83C23 C22⋊C4 C2×C4 C22 # reps 1 1 1 2 2 2 3 2 1 1 4 16 2

Matrix representation of C24.345C23 in GL6(𝔽5)

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

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

C24.345C23 in GAP, Magma, Sage, TeX

`C_2^4._{345}C_2^3`
`% in TeX`

`G:=Group("C2^4.345C2^3");`
`// GroupNames label`

`G:=SmallGroup(128,1319);`
`// by ID`

`G=gap.SmallGroup(128,1319);`
`# by ID`

`G:=PCGroup([7,-2,2,2,2,-2,2,2,253,792,758,723,352,675,136]);`
`// Polycyclic`

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