direct product, p-group, abelian, monomial
Aliases: C43, SmallGroup(64,55)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
| C1 — C43 |
| C1 — C43 |
| C1 — C43 |
Generators and relations for C43
G = < a,b,c | a4=b4=c4=1, ab=ba, ac=ca, bc=cb >
Subgroups: 129, all normal (3 characteristic)
C1, C2, C4, C22, C2×C4, C23, C42, C22×C4, C2×C42, C43
Quotients: C1, C2, C4, C22, C2×C4, C23, C42, C22×C4, C2×C42, C43
►Regular action on 64 points
(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 49 41 61)(2 50 42 62)(3 51 43 63)(4 52 44 64)(5 60 20 12)(6 57 17 9)(7 58 18 10)(8 59 19 11)(13 34 25 21)(14 35 26 22)(15 36 27 23)(16 33 28 24)(29 53 45 38)(30 54 46 39)(31 55 47 40)(32 56 48 37)
(1 47 11 25)(2 48 12 26)(3 45 9 27)(4 46 10 28)(5 22 50 37)(6 23 51 38)(7 24 52 39)(8 21 49 40)(13 41 31 59)(14 42 32 60)(15 43 29 57)(16 44 30 58)(17 36 63 53)(18 33 64 54)(19 34 61 55)(20 35 62 56)
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), (1,49,41,61)(2,50,42,62)(3,51,43,63)(4,52,44,64)(5,60,20,12)(6,57,17,9)(7,58,18,10)(8,59,19,11)(13,34,25,21)(14,35,26,22)(15,36,27,23)(16,33,28,24)(29,53,45,38)(30,54,46,39)(31,55,47,40)(32,56,48,37), (1,47,11,25)(2,48,12,26)(3,45,9,27)(4,46,10,28)(5,22,50,37)(6,23,51,38)(7,24,52,39)(8,21,49,40)(13,41,31,59)(14,42,32,60)(15,43,29,57)(16,44,30,58)(17,36,63,53)(18,33,64,54)(19,34,61,55)(20,35,62,56)>;
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), (1,49,41,61)(2,50,42,62)(3,51,43,63)(4,52,44,64)(5,60,20,12)(6,57,17,9)(7,58,18,10)(8,59,19,11)(13,34,25,21)(14,35,26,22)(15,36,27,23)(16,33,28,24)(29,53,45,38)(30,54,46,39)(31,55,47,40)(32,56,48,37), (1,47,11,25)(2,48,12,26)(3,45,9,27)(4,46,10,28)(5,22,50,37)(6,23,51,38)(7,24,52,39)(8,21,49,40)(13,41,31,59)(14,42,32,60)(15,43,29,57)(16,44,30,58)(17,36,63,53)(18,33,64,54)(19,34,61,55)(20,35,62,56) );
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)], [(1,49,41,61),(2,50,42,62),(3,51,43,63),(4,52,44,64),(5,60,20,12),(6,57,17,9),(7,58,18,10),(8,59,19,11),(13,34,25,21),(14,35,26,22),(15,36,27,23),(16,33,28,24),(29,53,45,38),(30,54,46,39),(31,55,47,40),(32,56,48,37)], [(1,47,11,25),(2,48,12,26),(3,45,9,27),(4,46,10,28),(5,22,50,37),(6,23,51,38),(7,24,52,39),(8,21,49,40),(13,41,31,59),(14,42,32,60),(15,43,29,57),(16,44,30,58),(17,36,63,53),(18,33,64,54),(19,34,61,55),(20,35,62,56)]])
C43 is a maximal subgroup of
C42⋊6C8 C42⋊4C8 C43.C2 C43.7C2 C42⋊8C8 C42⋊5C8 C42⋊9C8 C43⋊C2 C42⋊16Q8 C43⋊9C2 C42⋊14Q8 C43⋊2C2 C43⋊12C2 C43.15C2 C43⋊13C2 C43⋊14C2 C42⋊18Q8 C42⋊15Q8 C43.18C2 C43⋊4C2 C43⋊5C2 C43⋊15C2 C42⋊19Q8 C43⋊C7
C43 is a maximal quotient of
C2.C43
64 conjugacy classes
| class | 1 | 2A | ··· | 2G | 4A | ··· | 4BD |
| order | 1 | 2 | ··· | 2 | 4 | ··· | 4 |
| size | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
64 irreducible representations
| dim | 1 | 1 | 1 |
| type | + | + | |
| image | C1 | C2 | C4 |
| kernel | C43 | C2×C42 | C42 |
| # reps | 1 | 7 | 56 |
Matrix representation of C43 ►in GL3(𝔽5) generated by
| 3 | 0 | 0 |
| 0 | 4 | 0 |
| 0 | 0 | 2 |
| 3 | 0 | 0 |
| 0 | 2 | 0 |
| 0 | 0 | 1 |
| 4 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 3 |
G:=sub<GL(3,GF(5))| [3,0,0,0,4,0,0,0,2],[3,0,0,0,2,0,0,0,1],[4,0,0,0,1,0,0,0,3] >;
C43 in GAP, Magma, Sage, TeX
C_4^3
% in TeX
G:=Group("C4^3"); // GroupNames label
G:=SmallGroup(64,55);
// by ID
G=gap.SmallGroup(64,55);
# by ID
G:=PCGroup([6,-2,2,2,-2,2,2,48,103,158]);
// Polycyclic
G:=Group<a,b,c|a^4=b^4=c^4=1,a*b=b*a,a*c=c*a,b*c=c*b>;
// generators/relations