Finite Group DatabasebetaFinite Group Databasebeta| order | 1092 |
| isoclinism | isoclinic class of PSL(2,𝔽13) |
| automorphism | Aut(G) = PGL(2,𝔽13) (order 2184) · Inn(G) = G · Out(G) = C2 |
| homology | Abelianization (H1) = C1 · Schur multiplier (H2) = C2 |
| projective special linear | PSL(2,𝔽13) |
| projective symplectic | PSp(2,𝔽13) |
| projective special unitary | PSU(2,𝔽13) |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, hypocenter | G | C1 |
| LCS | G | |
| UCS | C1 | |
16 conjugacy classes of subgroups.
| minimal degree | μ(G) = 14 |
| transitive degree | μt(G) = 14 |
Permutation generators:
(1,2)(3,4)(5,8)(6,9)(7,12)(13,14);(2,3,5,9,6,11,8,13,10,7,12,14,4);(3,6,9,4,7,12)(5,10,8,14,11,13)| C1 | C2 | C3 | C4 | C5 | C6 | C7 | C8 | C9 | |
|---|---|---|---|---|---|---|---|---|---|
| size | 1 | 84 | 84 | 91 | 156 | 156 | 156 | 182 | 182 |
| order | 1 | 13 | 13 | 2 | 7 | 7 | 7 | 3 | 6 |
| χ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| χ2 | 7 | -1 | 0 | 0 | 0 | 1 | -1 | ||
| χ3 | 7 | -1 | 0 | 0 | 0 | 1 | -1 | ||
| χ4 | 12 | -1 | -1 | 0 | ζ2+ζ5+ζ9+ζ12+ζ16+ζ19+ζ23+ζ30+ζ33+ζ37+ζ40+ζ44+ζ47+ζ51+ζ54+ζ58+ζ61+ζ68+ζ72+ζ75+ζ79+ζ82+ζ86+ζ89 | ζ3+ζ4+ζ10+ζ11+ζ17+ζ18+ζ24+ζ25+ζ31+ζ32+ζ38+ζ45+ζ46+ζ53+ζ59+ζ60+ζ66+ζ67+ζ73+ζ74+ζ80+ζ81+ζ87+ζ88 | ζ+ζ6+ζ8+ζ15+ζ20+ζ22+ζ27+ζ29+ζ34+ζ36+ζ41+ζ43+ζ48+ζ50+ζ55+ζ57+ζ62+ζ64+ζ69+ζ71+ζ76+ζ83+ζ85+ζ90 | 0 | 0 |
| χ5 | 12 | -1 | -1 | 0 | ζ3+ζ4+ζ10+ζ11+ζ17+ζ18+ζ24+ζ25+ζ31+ζ32+ζ38+ζ45+ζ46+ζ53+ζ59+ζ60+ζ66+ζ67+ζ73+ζ74+ζ80+ζ81+ζ87+ζ88 | ζ+ζ6+ζ8+ζ15+ζ20+ζ22+ζ27+ζ29+ζ34+ζ36+ζ41+ζ43+ζ48+ζ50+ζ55+ζ57+ζ62+ζ64+ζ69+ζ71+ζ76+ζ83+ζ85+ζ90 | ζ2+ζ5+ζ9+ζ12+ζ16+ζ19+ζ23+ζ30+ζ33+ζ37+ζ40+ζ44+ζ47+ζ51+ζ54+ζ58+ζ61+ζ68+ζ72+ζ75+ζ79+ζ82+ζ86+ζ89 | 0 | 0 |
| χ6 | 12 | -1 | -1 | 0 | ζ+ζ6+ζ8+ζ15+ζ20+ζ22+ζ27+ζ29+ζ34+ζ36+ζ41+ζ43+ζ48+ζ50+ζ55+ζ57+ζ62+ζ64+ζ69+ζ71+ζ76+ζ83+ζ85+ζ90 | ζ2+ζ5+ζ9+ζ12+ζ16+ζ19+ζ23+ζ30+ζ33+ζ37+ζ40+ζ44+ζ47+ζ51+ζ54+ζ58+ζ61+ζ68+ζ72+ζ75+ζ79+ζ82+ζ86+ζ89 | ζ3+ζ4+ζ10+ζ11+ζ17+ζ18+ζ24+ζ25+ζ31+ζ32+ζ38+ζ45+ζ46+ζ53+ζ59+ζ60+ζ66+ζ67+ζ73+ζ74+ζ80+ζ81+ζ87+ζ88 | 0 | 0 |
| χ7 | 13 | 0 | 0 | 1 | -1 | -1 | -1 | 1 | 1 |
| χ8 | 14 | 1 | 1 | 2 | 0 | 0 | 0 | -1 | -1 |
| χ9 | 14 | 1 | 1 | -2 | 0 | 0 | 0 | -1 | 1 |
Here ζ is a primitive root of unity of order 91 (ζ91 = 1).
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1092, 25);