Finite Group DatabasebetaFinite Group Databasebeta| order | 31434 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | (C13⋊C3)⋊D403 |
| Fitting, socle, derived, Hall | C132×C31 ⋊ C2×C3 |
| Hall | C13×C31×C13⋊C3 ⋊ C2 |
| Hall | C13⋊D403 ⋊ C3 |
| Hall | C13×C13⋊C3 ⋊ D31 |
| Sylow | C132 ⋊ C3×D31 |
| Sylow | C31 ⋊ (C13⋊C3)⋊D13 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C132×C31 | C2×C3 |
| LCS | G ▷ C132×C31 | |
| UCS | C1 | |
48 conjugacy classes of subgroups.
| minimal degree | μ(G) = 57 |
| transitive degree | μt(G) = 1209 |
A minimal faithful permutation representation — 2 generators on 57 points:
(2,6,7,3,4,8)(5,12,11,9,10,13)(15,16)(17,18)(19,20)(21,22)(23,24)(25,26)(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);(1,2,5,4,8,11,10,12,13,7,6,9,3)(14,15,17,19,21,23,25,26,24,22,20,18,16)(27,28,30,32,34,36,38,40,42,44,46,48,50,52,54,56,57,55,53,51,49,47,45,43,41,39,37,35,33,31,29)Polycyclic presentation:
2,3,13^2,31|c^a=c^12,d^a=d^12,e^a=e^30,c^b=c^31415 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(31434, 43);