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## G = C23.67C23order 64 = 26

### 17th 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 — C23.67C23
 Chief series C1 — C2 — C22 — C23 — C22×C4 — C2×C42 — C23.67C23
 Lower central C1 — C22 — C23.67C23
 Upper central C1 — C23 — C23.67C23
 Jennings C1 — C23 — C23.67C23

Generators and relations for C23.67C23
G = < a,b,c,d,e,f | a2=b2=c2=1, d2=c, e2=abc, f2=b, 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: 137 in 93 conjugacy classes, 53 normal (13 characteristic)
C1, C2 [×3], C2 [×4], C4 [×4], C4 [×10], C22 [×3], C22 [×4], C2×C4 [×14], C2×C4 [×14], Q8 [×8], C23, C42 [×2], C4⋊C4 [×2], C22×C4, C22×C4 [×6], C2×Q8 [×4], C2×Q8 [×4], C2.C42 [×4], C2×C42, C2×C4⋊C4, C22×Q8, C23.67C23
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], Q8 [×4], C23, C22⋊C4 [×4], C22×C4, C2×D4 [×2], C2×Q8 [×2], C4○D4 [×2], C2×C22⋊C4, C4×Q8 [×2], C22⋊Q8 [×2], C4.4D4, C4⋊Q8, C23.67C23

Character table of C23.67C23

 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 2 0 0 -2 -2 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ18 2 -2 2 2 -2 -2 -2 2 2 0 0 2 -2 -2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ19 2 -2 2 2 -2 -2 -2 2 -2 0 0 -2 2 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ20 2 2 2 -2 -2 2 -2 -2 -2 0 0 2 2 -2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ21 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 symplectic lifted from Q8, Schur index 2 ρ22 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 ρ23 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 symplectic lifted from Q8, Schur index 2 ρ24 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 ρ25 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 ρ26 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 ρ27 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 ρ28 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

Smallest permutation representation of C23.67C23
Regular action on 64 points
Generators in S64
```(1 11)(2 12)(3 9)(4 10)(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 36)(6 33)(7 34)(8 35)(9 21)(10 22)(11 23)(12 24)(13 25)(14 26)(15 27)(16 28)(17 29)(18 30)(19 31)(20 32)(37 61)(38 62)(39 63)(40 64)(41 53)(42 54)(43 55)(44 56)(45 57)(46 58)(47 59)(48 60)
(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 29 21 47)(2 58 22 20)(3 31 23 45)(4 60 24 18)(5 42 64 28)(6 15 61 53)(7 44 62 26)(8 13 63 55)(9 59 51 17)(10 32 52 46)(11 57 49 19)(12 30 50 48)(14 34 56 38)(16 36 54 40)(25 39 43 35)(27 37 41 33)
(1 55 51 43)(2 44 52 56)(3 53 49 41)(4 42 50 54)(5 30 36 18)(6 19 33 31)(7 32 34 20)(8 17 35 29)(9 25 21 13)(10 14 22 26)(11 27 23 15)(12 16 24 28)(37 45 61 57)(38 58 62 46)(39 47 63 59)(40 60 64 48)```

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

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

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

Matrix representation of C23.67C23 in GL6(𝔽5)

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

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

C23.67C23 in GAP, Magma, Sage, TeX

`C_2^3._{67}C_2^3`
`% in TeX`

`G:=Group("C2^3.67C2^3");`
`// GroupNames label`

`G:=SmallGroup(64,72);`
`// by ID`

`G=gap.SmallGroup(64,72);`
`# by ID`

`G:=PCGroup([6,-2,2,2,-2,2,2,192,121,343,362,86]);`
`// Polycyclic`

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