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G = C26order 64 = 26

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

direct product, p-group, elementary abelian, monomial, rational

Aliases: C26, SmallGroup(64,267)

Series: Derived Chief Lower central Upper central Jennings

C1 — C26
C1C2C22C23C24C25 — C26
C1 — C26
C1 — C26
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 [×63], C22 [×651], C23 [×1395], C24 [×651], C25 [×63], C26
Quotients: C1, C2 [×63], C22 [×651], C23 [×1395], C24 [×651], C25 [×63], 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 11)(2 12)(3 41)(4 42)(5 13)(6 14)(7 15)(8 16)(9 17)(10 18)(19 27)(20 28)(21 29)(22 30)(23 31)(24 32)(25 34)(26 33)(35 43)(36 44)(37 45)(38 46)(39 47)(40 48)(49 57)(50 58)(51 59)(52 60)(53 61)(54 62)(55 63)(56 64)
(1 7)(2 8)(3 45)(4 46)(5 9)(6 10)(11 15)(12 16)(13 17)(14 18)(19 23)(20 24)(21 25)(22 26)(27 31)(28 32)(29 34)(30 33)(35 39)(36 40)(37 41)(38 42)(43 47)(44 48)(49 53)(50 54)(51 55)(52 56)(57 61)(58 62)(59 63)(60 64)
(1 5)(2 6)(3 47)(4 48)(7 9)(8 10)(11 13)(12 14)(15 17)(16 18)(19 21)(20 22)(23 25)(24 26)(27 29)(28 30)(31 34)(32 33)(35 37)(36 38)(39 41)(40 42)(43 45)(44 46)(49 51)(50 52)(53 55)(54 56)(57 59)(58 60)(61 63)(62 64)
(1 35)(2 36)(3 17)(4 18)(5 37)(6 38)(7 39)(8 40)(9 41)(10 42)(11 43)(12 44)(13 45)(14 46)(15 47)(16 48)(19 49)(20 50)(21 51)(22 52)(23 53)(24 54)(25 55)(26 56)(27 57)(28 58)(29 59)(30 60)(31 61)(32 62)(33 64)(34 63)
(1 49)(2 50)(3 34)(4 33)(5 51)(6 52)(7 53)(8 54)(9 55)(10 56)(11 57)(12 58)(13 59)(14 60)(15 61)(16 62)(17 63)(18 64)(19 35)(20 36)(21 37)(22 38)(23 39)(24 40)(25 41)(26 42)(27 43)(28 44)(29 45)(30 46)(31 47)(32 48)

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

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

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

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
order12···2
size11···1

64 irreducible representations

dim11
type++
imageC1C2
kernelC26C25
# reps163

Matrix representation of C26 in GL6(ℤ)

-100000
0-10000
001000
000100
0000-10
000001
,
100000
0-10000
001000
000-100
0000-10
00000-1
,
100000
010000
001000
000-100
0000-10
00000-1
,
100000
010000
001000
000-100
0000-10
000001
,
100000
010000
001000
000100
0000-10
000001
,
100000
0-10000
00-1000
000100
000010
000001

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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