Finite Group DatabasebetaFinite Group Databasebeta| order | 31434 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C13 |
| unambiguous semidirect product | (C31⋊C3)⋊(C13⋊D13) |
| Fitting, socle, derived, Hall | C132×C31 ⋊ C2×C3 |
| Hall | C132×C31⋊C3 ⋊ C2 |
| Hall | C13⋊D403 ⋊ C3 |
| Sylow | C132 ⋊ D31⋊C3 |
| Hall | C31⋊C3 ⋊ C13⋊D13 |
| Sylow | C31 ⋊ C3×C13⋊D13 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C132×C31 | C2×C3 |
| LCS | G ▷ C132×C31 | |
| UCS | C1 | |
128 conjugacy classes of subgroups.
| minimal degree | μ(G) = 57 |
| transitive degree | μt(G) = 5239 |
A minimal faithful permutation representation — 3 generators on 57 points:
(2,3)(4,5)(6,7)(8,9)(10,11)(12,13)(15,16)(17,18)(19,20)(21,22)(23,24)(25,26)(28,32,33,29,30,34)(31,40,41,35,36,48)(37,51,52,43,44,56)(38,50,46,45,42,39)(47,49,57,53,54,55);(14,15,17,19,21,23,25,26,24,22,20,18,16)(27,28,31,37,45,30,34,39,47,54,36,42,48,52,57,44,51,55,56,41,50,40,49,53,46,33,32,38,43,35,29);(1,2,4,6,8,10,12,13,11,9,7,5,3)Polycyclic presentation:
2,3,13^2,31|c^a=c^12,d^a=d^12,e^a=e^30,e^b=e^51103 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(31434, 79);