direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: D46, C2×D23, C46⋊C2, C23⋊C22, sometimes denoted D92 or Dih46 or Dih92, SmallGroup(92,3)
Series: Derived ►Chief ►Lower central ►Upper central
| C23 — D46 |
Generators and relations for D46
G = < a,b | a46=b2=1, bab=a-1 >
Character table of D46
| class | 1 | 2A | 2B | 2C | 23A | 23B | 23C | 23D | 23E | 23F | 23G | 23H | 23I | 23J | 23K | 46A | 46B | 46C | 46D | 46E | 46F | 46G | 46H | 46I | 46J | 46K | |
| size | 1 | 1 | 23 | 23 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | |
| 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 | 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 | 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 | 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 | linear of order 2 |
| ρ5 | 2 | 2 | 0 | 0 | ζ2315+ζ238 | ζ2313+ζ2310 | ζ2312+ζ2311 | ζ2314+ζ239 | ζ2316+ζ237 | ζ2318+ζ235 | ζ2320+ζ233 | ζ2322+ζ23 | ζ2321+ζ232 | ζ2319+ζ234 | ζ2317+ζ236 | ζ2317+ζ236 | ζ2315+ζ238 | ζ2313+ζ2310 | ζ2312+ζ2311 | ζ2314+ζ239 | ζ2316+ζ237 | ζ2318+ζ235 | ζ2320+ζ233 | ζ2322+ζ23 | ζ2321+ζ232 | ζ2319+ζ234 | orthogonal lifted from D23 |
| ρ6 | 2 | 2 | 0 | 0 | ζ2319+ζ234 | ζ2318+ζ235 | ζ2317+ζ236 | ζ2316+ζ237 | ζ2315+ζ238 | ζ2314+ζ239 | ζ2313+ζ2310 | ζ2312+ζ2311 | ζ2322+ζ23 | ζ2321+ζ232 | ζ2320+ζ233 | ζ2320+ζ233 | ζ2319+ζ234 | ζ2318+ζ235 | ζ2317+ζ236 | ζ2316+ζ237 | ζ2315+ζ238 | ζ2314+ζ239 | ζ2313+ζ2310 | ζ2312+ζ2311 | ζ2322+ζ23 | ζ2321+ζ232 | orthogonal lifted from D23 |
| ρ7 | 2 | -2 | 0 | 0 | ζ2320+ζ233 | ζ2321+ζ232 | ζ2316+ζ237 | ζ2312+ζ2311 | ζ2317+ζ236 | ζ2322+ζ23 | ζ2319+ζ234 | ζ2314+ζ239 | ζ2318+ζ235 | ζ2313+ζ2310 | ζ2315+ζ238 | -ζ2315-ζ238 | -ζ2320-ζ233 | -ζ2321-ζ232 | -ζ2316-ζ237 | -ζ2312-ζ2311 | -ζ2317-ζ236 | -ζ2322-ζ23 | -ζ2319-ζ234 | -ζ2314-ζ239 | -ζ2318-ζ235 | -ζ2313-ζ2310 | orthogonal faithful |
| ρ8 | 2 | -2 | 0 | 0 | ζ2321+ζ232 | ζ2314+ζ239 | ζ2320+ζ233 | ζ2315+ζ238 | ζ2319+ζ234 | ζ2316+ζ237 | ζ2318+ζ235 | ζ2317+ζ236 | ζ2312+ζ2311 | ζ2322+ζ23 | ζ2313+ζ2310 | -ζ2313-ζ2310 | -ζ2321-ζ232 | -ζ2314-ζ239 | -ζ2320-ζ233 | -ζ2315-ζ238 | -ζ2319-ζ234 | -ζ2316-ζ237 | -ζ2318-ζ235 | -ζ2317-ζ236 | -ζ2312-ζ2311 | -ζ2322-ζ23 | orthogonal faithful |
| ρ9 | 2 | -2 | 0 | 0 | ζ2314+ζ239 | ζ2317+ζ236 | ζ2321+ζ232 | ζ2313+ζ2310 | ζ2318+ζ235 | ζ2320+ζ233 | ζ2312+ζ2311 | ζ2319+ζ234 | ζ2315+ζ238 | ζ2316+ζ237 | ζ2322+ζ23 | -ζ2322-ζ23 | -ζ2314-ζ239 | -ζ2317-ζ236 | -ζ2321-ζ232 | -ζ2313-ζ2310 | -ζ2318-ζ235 | -ζ2320-ζ233 | -ζ2312-ζ2311 | -ζ2319-ζ234 | -ζ2315-ζ238 | -ζ2316-ζ237 | orthogonal faithful |
| ρ10 | 2 | 2 | 0 | 0 | ζ2320+ζ233 | ζ2321+ζ232 | ζ2316+ζ237 | ζ2312+ζ2311 | ζ2317+ζ236 | ζ2322+ζ23 | ζ2319+ζ234 | ζ2314+ζ239 | ζ2318+ζ235 | ζ2313+ζ2310 | ζ2315+ζ238 | ζ2315+ζ238 | ζ2320+ζ233 | ζ2321+ζ232 | ζ2316+ζ237 | ζ2312+ζ2311 | ζ2317+ζ236 | ζ2322+ζ23 | ζ2319+ζ234 | ζ2314+ζ239 | ζ2318+ζ235 | ζ2313+ζ2310 | orthogonal lifted from D23 |
| ρ11 | 2 | 2 | 0 | 0 | ζ2316+ζ237 | ζ2320+ζ233 | ζ2322+ζ23 | ζ2318+ζ235 | ζ2314+ζ239 | ζ2313+ζ2310 | ζ2317+ζ236 | ζ2321+ζ232 | ζ2319+ζ234 | ζ2315+ζ238 | ζ2312+ζ2311 | ζ2312+ζ2311 | ζ2316+ζ237 | ζ2320+ζ233 | ζ2322+ζ23 | ζ2318+ζ235 | ζ2314+ζ239 | ζ2313+ζ2310 | ζ2317+ζ236 | ζ2321+ζ232 | ζ2319+ζ234 | ζ2315+ζ238 | orthogonal lifted from D23 |
| ρ12 | 2 | -2 | 0 | 0 | ζ2315+ζ238 | ζ2313+ζ2310 | ζ2312+ζ2311 | ζ2314+ζ239 | ζ2316+ζ237 | ζ2318+ζ235 | ζ2320+ζ233 | ζ2322+ζ23 | ζ2321+ζ232 | ζ2319+ζ234 | ζ2317+ζ236 | -ζ2317-ζ236 | -ζ2315-ζ238 | -ζ2313-ζ2310 | -ζ2312-ζ2311 | -ζ2314-ζ239 | -ζ2316-ζ237 | -ζ2318-ζ235 | -ζ2320-ζ233 | -ζ2322-ζ23 | -ζ2321-ζ232 | -ζ2319-ζ234 | orthogonal faithful |
| ρ13 | 2 | -2 | 0 | 0 | ζ2317+ζ236 | ζ2319+ζ234 | ζ2314+ζ239 | ζ2322+ζ23 | ζ2312+ζ2311 | ζ2321+ζ232 | ζ2315+ζ238 | ζ2318+ζ235 | ζ2313+ζ2310 | ζ2320+ζ233 | ζ2316+ζ237 | -ζ2316-ζ237 | -ζ2317-ζ236 | -ζ2319-ζ234 | -ζ2314-ζ239 | -ζ2322-ζ23 | -ζ2312-ζ2311 | -ζ2321-ζ232 | -ζ2315-ζ238 | -ζ2318-ζ235 | -ζ2313-ζ2310 | -ζ2320-ζ233 | orthogonal faithful |
| ρ14 | 2 | -2 | 0 | 0 | ζ2322+ζ23 | ζ2316+ζ237 | ζ2313+ζ2310 | ζ2319+ζ234 | ζ2321+ζ232 | ζ2315+ζ238 | ζ2314+ζ239 | ζ2320+ζ233 | ζ2317+ζ236 | ζ2312+ζ2311 | ζ2318+ζ235 | -ζ2318-ζ235 | -ζ2322-ζ23 | -ζ2316-ζ237 | -ζ2313-ζ2310 | -ζ2319-ζ234 | -ζ2321-ζ232 | -ζ2315-ζ238 | -ζ2314-ζ239 | -ζ2320-ζ233 | -ζ2317-ζ236 | -ζ2312-ζ2311 | orthogonal faithful |
| ρ15 | 2 | -2 | 0 | 0 | ζ2318+ζ235 | ζ2312+ζ2311 | ζ2319+ζ234 | ζ2320+ζ233 | ζ2313+ζ2310 | ζ2317+ζ236 | ζ2322+ζ23 | ζ2315+ζ238 | ζ2316+ζ237 | ζ2314+ζ239 | ζ2321+ζ232 | -ζ2321-ζ232 | -ζ2318-ζ235 | -ζ2312-ζ2311 | -ζ2319-ζ234 | -ζ2320-ζ233 | -ζ2313-ζ2310 | -ζ2317-ζ236 | -ζ2322-ζ23 | -ζ2315-ζ238 | -ζ2316-ζ237 | -ζ2314-ζ239 | orthogonal faithful |
| ρ16 | 2 | -2 | 0 | 0 | ζ2319+ζ234 | ζ2318+ζ235 | ζ2317+ζ236 | ζ2316+ζ237 | ζ2315+ζ238 | ζ2314+ζ239 | ζ2313+ζ2310 | ζ2312+ζ2311 | ζ2322+ζ23 | ζ2321+ζ232 | ζ2320+ζ233 | -ζ2320-ζ233 | -ζ2319-ζ234 | -ζ2318-ζ235 | -ζ2317-ζ236 | -ζ2316-ζ237 | -ζ2315-ζ238 | -ζ2314-ζ239 | -ζ2313-ζ2310 | -ζ2312-ζ2311 | -ζ2322-ζ23 | -ζ2321-ζ232 | orthogonal faithful |
| ρ17 | 2 | 2 | 0 | 0 | ζ2313+ζ2310 | ζ2322+ζ23 | ζ2315+ζ238 | ζ2317+ζ236 | ζ2320+ζ233 | ζ2312+ζ2311 | ζ2321+ζ232 | ζ2316+ζ237 | ζ2314+ζ239 | ζ2318+ζ235 | ζ2319+ζ234 | ζ2319+ζ234 | ζ2313+ζ2310 | ζ2322+ζ23 | ζ2315+ζ238 | ζ2317+ζ236 | ζ2320+ζ233 | ζ2312+ζ2311 | ζ2321+ζ232 | ζ2316+ζ237 | ζ2314+ζ239 | ζ2318+ζ235 | orthogonal lifted from D23 |
| ρ18 | 2 | -2 | 0 | 0 | ζ2316+ζ237 | ζ2320+ζ233 | ζ2322+ζ23 | ζ2318+ζ235 | ζ2314+ζ239 | ζ2313+ζ2310 | ζ2317+ζ236 | ζ2321+ζ232 | ζ2319+ζ234 | ζ2315+ζ238 | ζ2312+ζ2311 | -ζ2312-ζ2311 | -ζ2316-ζ237 | -ζ2320-ζ233 | -ζ2322-ζ23 | -ζ2318-ζ235 | -ζ2314-ζ239 | -ζ2313-ζ2310 | -ζ2317-ζ236 | -ζ2321-ζ232 | -ζ2319-ζ234 | -ζ2315-ζ238 | orthogonal faithful |
| ρ19 | 2 | 2 | 0 | 0 | ζ2318+ζ235 | ζ2312+ζ2311 | ζ2319+ζ234 | ζ2320+ζ233 | ζ2313+ζ2310 | ζ2317+ζ236 | ζ2322+ζ23 | ζ2315+ζ238 | ζ2316+ζ237 | ζ2314+ζ239 | ζ2321+ζ232 | ζ2321+ζ232 | ζ2318+ζ235 | ζ2312+ζ2311 | ζ2319+ζ234 | ζ2320+ζ233 | ζ2313+ζ2310 | ζ2317+ζ236 | ζ2322+ζ23 | ζ2315+ζ238 | ζ2316+ζ237 | ζ2314+ζ239 | orthogonal lifted from D23 |
| ρ20 | 2 | 2 | 0 | 0 | ζ2317+ζ236 | ζ2319+ζ234 | ζ2314+ζ239 | ζ2322+ζ23 | ζ2312+ζ2311 | ζ2321+ζ232 | ζ2315+ζ238 | ζ2318+ζ235 | ζ2313+ζ2310 | ζ2320+ζ233 | ζ2316+ζ237 | ζ2316+ζ237 | ζ2317+ζ236 | ζ2319+ζ234 | ζ2314+ζ239 | ζ2322+ζ23 | ζ2312+ζ2311 | ζ2321+ζ232 | ζ2315+ζ238 | ζ2318+ζ235 | ζ2313+ζ2310 | ζ2320+ζ233 | orthogonal lifted from D23 |
| ρ21 | 2 | 2 | 0 | 0 | ζ2314+ζ239 | ζ2317+ζ236 | ζ2321+ζ232 | ζ2313+ζ2310 | ζ2318+ζ235 | ζ2320+ζ233 | ζ2312+ζ2311 | ζ2319+ζ234 | ζ2315+ζ238 | ζ2316+ζ237 | ζ2322+ζ23 | ζ2322+ζ23 | ζ2314+ζ239 | ζ2317+ζ236 | ζ2321+ζ232 | ζ2313+ζ2310 | ζ2318+ζ235 | ζ2320+ζ233 | ζ2312+ζ2311 | ζ2319+ζ234 | ζ2315+ζ238 | ζ2316+ζ237 | orthogonal lifted from D23 |
| ρ22 | 2 | 2 | 0 | 0 | ζ2312+ζ2311 | ζ2315+ζ238 | ζ2318+ζ235 | ζ2321+ζ232 | ζ2322+ζ23 | ζ2319+ζ234 | ζ2316+ζ237 | ζ2313+ζ2310 | ζ2320+ζ233 | ζ2317+ζ236 | ζ2314+ζ239 | ζ2314+ζ239 | ζ2312+ζ2311 | ζ2315+ζ238 | ζ2318+ζ235 | ζ2321+ζ232 | ζ2322+ζ23 | ζ2319+ζ234 | ζ2316+ζ237 | ζ2313+ζ2310 | ζ2320+ζ233 | ζ2317+ζ236 | orthogonal lifted from D23 |
| ρ23 | 2 | -2 | 0 | 0 | ζ2312+ζ2311 | ζ2315+ζ238 | ζ2318+ζ235 | ζ2321+ζ232 | ζ2322+ζ23 | ζ2319+ζ234 | ζ2316+ζ237 | ζ2313+ζ2310 | ζ2320+ζ233 | ζ2317+ζ236 | ζ2314+ζ239 | -ζ2314-ζ239 | -ζ2312-ζ2311 | -ζ2315-ζ238 | -ζ2318-ζ235 | -ζ2321-ζ232 | -ζ2322-ζ23 | -ζ2319-ζ234 | -ζ2316-ζ237 | -ζ2313-ζ2310 | -ζ2320-ζ233 | -ζ2317-ζ236 | orthogonal faithful |
| ρ24 | 2 | -2 | 0 | 0 | ζ2313+ζ2310 | ζ2322+ζ23 | ζ2315+ζ238 | ζ2317+ζ236 | ζ2320+ζ233 | ζ2312+ζ2311 | ζ2321+ζ232 | ζ2316+ζ237 | ζ2314+ζ239 | ζ2318+ζ235 | ζ2319+ζ234 | -ζ2319-ζ234 | -ζ2313-ζ2310 | -ζ2322-ζ23 | -ζ2315-ζ238 | -ζ2317-ζ236 | -ζ2320-ζ233 | -ζ2312-ζ2311 | -ζ2321-ζ232 | -ζ2316-ζ237 | -ζ2314-ζ239 | -ζ2318-ζ235 | orthogonal faithful |
| ρ25 | 2 | 2 | 0 | 0 | ζ2321+ζ232 | ζ2314+ζ239 | ζ2320+ζ233 | ζ2315+ζ238 | ζ2319+ζ234 | ζ2316+ζ237 | ζ2318+ζ235 | ζ2317+ζ236 | ζ2312+ζ2311 | ζ2322+ζ23 | ζ2313+ζ2310 | ζ2313+ζ2310 | ζ2321+ζ232 | ζ2314+ζ239 | ζ2320+ζ233 | ζ2315+ζ238 | ζ2319+ζ234 | ζ2316+ζ237 | ζ2318+ζ235 | ζ2317+ζ236 | ζ2312+ζ2311 | ζ2322+ζ23 | orthogonal lifted from D23 |
| ρ26 | 2 | 2 | 0 | 0 | ζ2322+ζ23 | ζ2316+ζ237 | ζ2313+ζ2310 | ζ2319+ζ234 | ζ2321+ζ232 | ζ2315+ζ238 | ζ2314+ζ239 | ζ2320+ζ233 | ζ2317+ζ236 | ζ2312+ζ2311 | ζ2318+ζ235 | ζ2318+ζ235 | ζ2322+ζ23 | ζ2316+ζ237 | ζ2313+ζ2310 | ζ2319+ζ234 | ζ2321+ζ232 | ζ2315+ζ238 | ζ2314+ζ239 | ζ2320+ζ233 | ζ2317+ζ236 | ζ2312+ζ2311 | orthogonal lifted from D23 |
►On 46 points
Generators in S46
(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)
(1 46)(2 45)(3 44)(4 43)(5 42)(6 41)(7 40)(8 39)(9 38)(10 37)(11 36)(12 35)(13 34)(14 33)(15 32)(16 31)(17 30)(18 29)(19 28)(20 27)(21 26)(22 25)(23 24)
G:=sub<Sym(46)| (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), (1,46)(2,45)(3,44)(4,43)(5,42)(6,41)(7,40)(8,39)(9,38)(10,37)(11,36)(12,35)(13,34)(14,33)(15,32)(16,31)(17,30)(18,29)(19,28)(20,27)(21,26)(22,25)(23,24)>;
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), (1,46)(2,45)(3,44)(4,43)(5,42)(6,41)(7,40)(8,39)(9,38)(10,37)(11,36)(12,35)(13,34)(14,33)(15,32)(16,31)(17,30)(18,29)(19,28)(20,27)(21,26)(22,25)(23,24) );
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)], [(1,46),(2,45),(3,44),(4,43),(5,42),(6,41),(7,40),(8,39),(9,38),(10,37),(11,36),(12,35),(13,34),(14,33),(15,32),(16,31),(17,30),(18,29),(19,28),(20,27),(21,26),(22,25),(23,24)]])
D46 is a maximal subgroup of
D92 C23⋊D4
D46 is a maximal quotient of Dic46 D92 C23⋊D4
Matrix representation of D46 ►in GL2(𝔽47) generated by
| 0 | 46 |
| 1 | 13 |
| 13 | 27 |
| 46 | 34 |
G:=sub<GL(2,GF(47))| [0,1,46,13],[13,46,27,34] >;
D46 in GAP, Magma, Sage, TeX
D_{46} % in TeX
G:=Group("D46"); // GroupNames label
G:=SmallGroup(92,3);
// by ID
G=gap.SmallGroup(92,3);
# by ID
G:=PCGroup([3,-2,-2,-23,794]);
// Polycyclic
G:=Group<a,b|a^46=b^2=1,b*a*b=a^-1>;
// generators/relations
Export
Subgroup lattice of D46 in TeX
Character table of D46 in TeX