direct product, abelian, monomial, 2-elementary
Aliases: C23×C6, SmallGroup(48,52)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C23×C6 |
| C1 — C23×C6 |
| C1 — C23×C6 |
Generators and relations for C23×C6
G = < a,b,c,d | a2=b2=c2=d6=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, cd=dc >
Subgroups: 134, all normal (4 characteristic)
C1, C2, C3, C22, C6, C23, C2×C6, C24, C22×C6, C23×C6
Quotients: C1, C2, C3, C22, C6, C23, C2×C6, C24, C22×C6, C23×C6
►Regular action on 48 points
Generators in S48
(1 29)(2 30)(3 25)(4 26)(5 27)(6 28)(7 31)(8 32)(9 33)(10 34)(11 35)(12 36)(13 37)(14 38)(15 39)(16 40)(17 41)(18 42)(19 43)(20 44)(21 45)(22 46)(23 47)(24 48)
(1 17)(2 18)(3 13)(4 14)(5 15)(6 16)(7 19)(8 20)(9 21)(10 22)(11 23)(12 24)(25 37)(26 38)(27 39)(28 40)(29 41)(30 42)(31 43)(32 44)(33 45)(34 46)(35 47)(36 48)
(1 11)(2 12)(3 7)(4 8)(5 9)(6 10)(13 19)(14 20)(15 21)(16 22)(17 23)(18 24)(25 31)(26 32)(27 33)(28 34)(29 35)(30 36)(37 43)(38 44)(39 45)(40 46)(41 47)(42 48)
(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)
G:=sub<Sym(48)| (1,29)(2,30)(3,25)(4,26)(5,27)(6,28)(7,31)(8,32)(9,33)(10,34)(11,35)(12,36)(13,37)(14,38)(15,39)(16,40)(17,41)(18,42)(19,43)(20,44)(21,45)(22,46)(23,47)(24,48), (1,17)(2,18)(3,13)(4,14)(5,15)(6,16)(7,19)(8,20)(9,21)(10,22)(11,23)(12,24)(25,37)(26,38)(27,39)(28,40)(29,41)(30,42)(31,43)(32,44)(33,45)(34,46)(35,47)(36,48), (1,11)(2,12)(3,7)(4,8)(5,9)(6,10)(13,19)(14,20)(15,21)(16,22)(17,23)(18,24)(25,31)(26,32)(27,33)(28,34)(29,35)(30,36)(37,43)(38,44)(39,45)(40,46)(41,47)(42,48), (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)>;
G:=Group( (1,29)(2,30)(3,25)(4,26)(5,27)(6,28)(7,31)(8,32)(9,33)(10,34)(11,35)(12,36)(13,37)(14,38)(15,39)(16,40)(17,41)(18,42)(19,43)(20,44)(21,45)(22,46)(23,47)(24,48), (1,17)(2,18)(3,13)(4,14)(5,15)(6,16)(7,19)(8,20)(9,21)(10,22)(11,23)(12,24)(25,37)(26,38)(27,39)(28,40)(29,41)(30,42)(31,43)(32,44)(33,45)(34,46)(35,47)(36,48), (1,11)(2,12)(3,7)(4,8)(5,9)(6,10)(13,19)(14,20)(15,21)(16,22)(17,23)(18,24)(25,31)(26,32)(27,33)(28,34)(29,35)(30,36)(37,43)(38,44)(39,45)(40,46)(41,47)(42,48), (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) );
G=PermutationGroup([[(1,29),(2,30),(3,25),(4,26),(5,27),(6,28),(7,31),(8,32),(9,33),(10,34),(11,35),(12,36),(13,37),(14,38),(15,39),(16,40),(17,41),(18,42),(19,43),(20,44),(21,45),(22,46),(23,47),(24,48)], [(1,17),(2,18),(3,13),(4,14),(5,15),(6,16),(7,19),(8,20),(9,21),(10,22),(11,23),(12,24),(25,37),(26,38),(27,39),(28,40),(29,41),(30,42),(31,43),(32,44),(33,45),(34,46),(35,47),(36,48)], [(1,11),(2,12),(3,7),(4,8),(5,9),(6,10),(13,19),(14,20),(15,21),(16,22),(17,23),(18,24),(25,31),(26,32),(27,33),(28,34),(29,35),(30,36),(37,43),(38,44),(39,45),(40,46),(41,47),(42,48)], [(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)]])
C23×C6 is a maximal subgroup of
C24⋊4S3 C24⋊C9
48 conjugacy classes
| class | 1 | 2A | ··· | 2O | 3A | 3B | 6A | ··· | 6AD |
| order | 1 | 2 | ··· | 2 | 3 | 3 | 6 | ··· | 6 |
| size | 1 | 1 | ··· | 1 | 1 | 1 | 1 | ··· | 1 |
48 irreducible representations
| dim | 1 | 1 | 1 | 1 |
| type | + | + | ||
| image | C1 | C2 | C3 | C6 |
| kernel | C23×C6 | C22×C6 | C24 | C23 |
| # reps | 1 | 15 | 2 | 30 |
Matrix representation of C23×C6 ►in GL4(𝔽7) generated by
| 6 | 0 | 0 | 0 |
| 0 | 6 | 0 | 0 |
| 0 | 0 | 6 | 0 |
| 0 | 0 | 0 | 6 |
| 6 | 0 | 0 | 0 |
| 0 | 6 | 0 | 0 |
| 0 | 0 | 6 | 0 |
| 0 | 0 | 0 | 1 |
| 6 | 0 | 0 | 0 |
| 0 | 6 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 6 |
| 5 | 0 | 0 | 0 |
| 0 | 2 | 0 | 0 |
| 0 | 0 | 2 | 0 |
| 0 | 0 | 0 | 5 |
G:=sub<GL(4,GF(7))| [6,0,0,0,0,6,0,0,0,0,6,0,0,0,0,6],[6,0,0,0,0,6,0,0,0,0,6,0,0,0,0,1],[6,0,0,0,0,6,0,0,0,0,1,0,0,0,0,6],[5,0,0,0,0,2,0,0,0,0,2,0,0,0,0,5] >;
C23×C6 in GAP, Magma, Sage, TeX
C_2^3\times C_6
% in TeX
G:=Group("C2^3xC6"); // GroupNames label
G:=SmallGroup(48,52);
// by ID
G=gap.SmallGroup(48,52);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-3]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^2=c^2=d^6=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,c*d=d*c>;
// generators/relations