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## G = C23.63C23order 64 = 26

### 13rd central extension by C23 of C23

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C22 — C23.63C23
 Chief series C1 — C2 — C22 — C23 — C22×C4 — C2×C42 — C23.63C23
 Lower central C1 — C22 — C23.63C23
 Upper central C1 — C23 — C23.63C23
 Jennings C1 — C23 — C23.63C23

Generators and relations for C23.63C23
G = < a,b,c,d,e,f | a2=b2=c2=1, d2=e2=c, f2=a, ab=ba, ac=ca, ede-1=ad=da, ae=ea, af=fa, bc=cb, fdf-1=bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, ef=fe >

Subgroups: 113 in 77 conjugacy classes, 45 normal (31 characteristic)
C1, C2 [×7], C4 [×12], C22 [×7], C2×C4 [×10], C2×C4 [×16], C23, C42 [×2], C4⋊C4 [×4], C4⋊C4 [×2], C22×C4 [×7], C2.C42 [×4], C2×C42, C2×C4⋊C4 [×2], C23.63C23
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], Q8 [×2], C23, C22×C4, C2×D4, C2×Q8, C4○D4 [×4], C42⋊C2, C4×D4, C4×Q8, C22⋊Q8, C22.D4, C42.C2, C422C2, C23.63C23

Character table of C23.63C23

 class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 4M 4N 4O 4P 4Q 4R 4S 4T size 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 1 1 1 1 1 1 -1 -1 1 -1 -1 -1 -1 -1 -1 1 1 1 -1 -1 1 1 1 -1 -1 1 linear of order 2 ρ3 1 1 1 1 1 1 1 1 -1 1 -1 -1 -1 -1 1 1 1 -1 -1 -1 -1 1 -1 1 1 -1 1 -1 linear of order 2 ρ4 1 1 1 1 1 1 1 1 1 -1 -1 1 1 1 -1 -1 -1 -1 -1 -1 1 -1 -1 1 1 1 -1 -1 linear of order 2 ρ5 1 1 1 1 1 1 1 1 -1 1 -1 -1 -1 -1 1 1 1 -1 -1 -1 1 -1 1 -1 -1 1 -1 1 linear of order 2 ρ6 1 1 1 1 1 1 1 1 -1 -1 1 -1 -1 -1 -1 -1 -1 1 1 1 1 1 -1 -1 -1 1 1 -1 linear of order 2 ρ7 1 1 1 1 1 1 1 1 1 -1 -1 1 1 1 -1 -1 -1 -1 -1 -1 -1 1 1 -1 -1 -1 1 1 linear of order 2 ρ8 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 linear of order 2 ρ9 1 -1 1 -1 1 -1 1 -1 -1 i -i 1 -1 1 -i i -i i -i i -i -1 1 i -i i 1 -1 linear of order 4 ρ10 1 -1 1 -1 1 -1 1 -1 1 -i -i -1 1 -1 i -i i i -i i i 1 1 i -i -i -1 -1 linear of order 4 ρ11 1 -1 1 -1 1 -1 1 -1 -1 i -i 1 -1 1 -i i -i i -i i i 1 -1 -i i -i -1 1 linear of order 4 ρ12 1 -1 1 -1 1 -1 1 -1 1 -i -i -1 1 -1 i -i i i -i i -i -1 -1 -i i i 1 1 linear of order 4 ρ13 1 -1 1 -1 1 -1 1 -1 1 i i -1 1 -1 -i i -i -i i -i i -1 -1 i -i -i 1 1 linear of order 4 ρ14 1 -1 1 -1 1 -1 1 -1 -1 -i i 1 -1 1 i -i i -i i -i -i 1 -1 i -i i -1 1 linear of order 4 ρ15 1 -1 1 -1 1 -1 1 -1 1 i i -1 1 -1 -i i -i -i i -i -i 1 1 -i i i -1 -1 linear of order 4 ρ16 1 -1 1 -1 1 -1 1 -1 -1 -i i 1 -1 1 i -i i -i i -i i -1 1 -i i -i 1 -1 linear of order 4 ρ17 2 -2 -2 -2 2 2 -2 2 0 2 0 0 0 0 2 -2 -2 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ18 2 -2 -2 -2 2 2 -2 2 0 -2 0 0 0 0 -2 2 2 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ19 2 -2 -2 2 -2 2 2 -2 0 0 -2 0 0 0 0 0 0 2 2 -2 0 0 0 0 0 0 0 0 symplectic lifted from Q8, Schur index 2 ρ20 2 -2 -2 2 -2 2 2 -2 0 0 2 0 0 0 0 0 0 -2 -2 2 0 0 0 0 0 0 0 0 symplectic lifted from Q8, Schur index 2 ρ21 2 2 -2 2 2 -2 -2 -2 0 2i 0 0 0 0 -2i -2i 2i 0 0 0 0 0 0 0 0 0 0 0 complex lifted from C4○D4 ρ22 2 2 -2 2 2 -2 -2 -2 0 -2i 0 0 0 0 2i 2i -2i 0 0 0 0 0 0 0 0 0 0 0 complex lifted from C4○D4 ρ23 2 2 2 -2 -2 2 -2 -2 2i 0 0 -2i -2i 2i 0 0 0 0 0 0 0 0 0 0 0 0 0 0 complex lifted from C4○D4 ρ24 2 2 -2 -2 -2 -2 2 2 0 0 2i 0 0 0 0 0 0 2i -2i -2i 0 0 0 0 0 0 0 0 complex lifted from C4○D4 ρ25 2 -2 2 2 -2 -2 -2 2 2i 0 0 2i -2i -2i 0 0 0 0 0 0 0 0 0 0 0 0 0 0 complex lifted from C4○D4 ρ26 2 2 -2 -2 -2 -2 2 2 0 0 -2i 0 0 0 0 0 0 -2i 2i 2i 0 0 0 0 0 0 0 0 complex lifted from C4○D4 ρ27 2 -2 2 2 -2 -2 -2 2 -2i 0 0 -2i 2i 2i 0 0 0 0 0 0 0 0 0 0 0 0 0 0 complex lifted from C4○D4 ρ28 2 2 2 -2 -2 2 -2 -2 -2i 0 0 2i 2i -2i 0 0 0 0 0 0 0 0 0 0 0 0 0 0 complex lifted from C4○D4

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

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

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

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

Matrix representation of C23.63C23 in GL5(𝔽5)

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

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

C23.63C23 in GAP, Magma, Sage, TeX

`C_2^3._{63}C_2^3`
`% in TeX`

`G:=Group("C2^3.63C2^3");`
`// GroupNames label`

`G:=SmallGroup(64,68);`
`// by ID`

`G=gap.SmallGroup(64,68);`
`# by ID`

`G:=PCGroup([6,-2,2,2,-2,2,2,192,121,199,362,50]);`
`// Polycyclic`

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

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