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## G = C26order 64 = 26

### Elementary abelian group of type [2,2,2,2,2,2]

Aliases: C26, SmallGroup(64,267)

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C26
 Chief series C1 — C2 — C22 — C23 — C24 — C25 — C26
 Lower central C1 — C26
 Upper central C1 — C26
 Jennings C1 — C26

Generators and relations for C26
G = < a,b,c,d,e,f | a2=b2=c2=d2=e2=f2=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, ef=fe >

Subgroups: 2825, all normal (2 characteristic)
C1, C2, C22, C23, C24, C25, C26
Quotients: C1, C2, C22, C23, C24, C25, C26

Smallest permutation representation of C26
Regular action on 64 points
Generators in S64
```(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 54)(2 53)(3 48)(4 47)(5 15)(6 16)(7 31)(8 32)(9 42)(10 41)(11 35)(12 36)(13 21)(14 22)(17 23)(18 24)(19 40)(20 39)(25 60)(26 59)(27 62)(28 61)(29 44)(30 43)(33 50)(34 49)(37 57)(38 58)(45 63)(46 64)(51 55)(52 56)
(1 37)(2 38)(3 46)(4 45)(5 13)(6 14)(7 43)(8 44)(9 34)(10 33)(11 25)(12 26)(15 21)(16 22)(17 19)(18 20)(23 40)(24 39)(27 56)(28 55)(29 32)(30 31)(35 60)(36 59)(41 50)(42 49)(47 63)(48 64)(51 61)(52 62)(53 58)(54 57)
(1 41)(2 42)(3 32)(4 31)(5 23)(6 24)(7 47)(8 48)(9 53)(10 54)(11 61)(12 62)(13 40)(14 39)(15 17)(16 18)(19 21)(20 22)(25 51)(26 52)(27 36)(28 35)(29 46)(30 45)(33 57)(34 58)(37 50)(38 49)(43 63)(44 64)(55 60)(56 59)
(1 7)(2 8)(3 9)(4 10)(5 59)(6 60)(11 22)(12 21)(13 36)(14 35)(15 26)(16 25)(17 52)(18 51)(19 62)(20 61)(23 56)(24 55)(27 40)(28 39)(29 58)(30 57)(31 54)(32 53)(33 45)(34 46)(37 43)(38 44)(41 47)(42 48)(49 64)(50 63)
(1 21)(2 22)(3 28)(4 27)(5 57)(6 58)(7 12)(8 11)(9 39)(10 40)(13 54)(14 53)(15 37)(16 38)(17 50)(18 49)(19 41)(20 42)(23 33)(24 34)(25 44)(26 43)(29 60)(30 59)(31 36)(32 35)(45 56)(46 55)(47 62)(48 61)(51 64)(52 63)```

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

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

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

C26 is a maximal subgroup of   C23≀C2  C26⋊C3  C26⋊C7  C23⋊F8
C26 is a maximal quotient of   2+ 1+6  2- 1+6

64 conjugacy classes

 class 1 2A ··· 2BK order 1 2 ··· 2 size 1 1 ··· 1

64 irreducible representations

 dim 1 1 type + + image C1 C2 kernel C26 C25 # reps 1 63

Matrix representation of C26 in GL6(ℤ)

 -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 -1 0 0 0 0 0 0 1
,
 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 -1 0 0 0 0 0 0 -1
,
 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 -1 0 0 0 0 0 0 -1
,
 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 -1 0 0 0 0 0 0 1
,
 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 -1 0 0 0 0 0 0 1
,
 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 1 0 0 0 0 0 0 1

`G:=sub<GL(6,Integers())| [-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,-1,0,0,0,0,0,0,1],[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,-1,0,0,0,0,0,0,-1],[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,-1,0,0,0,0,0,0,-1],[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,-1,0,0,0,0,0,0,1],[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,-1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;`

C26 in GAP, Magma, Sage, TeX

`C_2^6`
`% in TeX`

`G:=Group("C2^6");`
`// GroupNames label`

`G:=SmallGroup(64,267);`
`// by ID`

`G=gap.SmallGroup(64,267);`
`# by ID`

`G:=PCGroup([6,-2,2,2,2,2,2]);`
`// Polycyclic`

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

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