Finite Group DatabasebetaFinite Group Databasebeta| order | 399 |
| isoclinism | isoclinic class of #87704 |
| automorphism | ? |
| homology | Abelianization (H1) = C3 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Fitting, socle, derived, Hall | C7×C19 ⋊ C3 |
| Sylow | C19 ⋊ ΓL(1,𝔽8) |
| Sylow | C7 ⋊ C19⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C7×C19 | C3 |
| LCS | G ▷ C7×C19 | |
| UCS | C1 | |
8 conjugacy classes of subgroups.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 133 |
A minimal faithful permutation representation — 2 generators on 26 points:
(2,3,5)(4,7,8)(6,10,9)(11,15,13)(12,16,14)(17,19,18)(21,22,23)(24,26,25);(1,2,4,8,13,18,15,3,6,11,16,5,9,14,7,12,17,19,10)(20,21,22,24,23,25,26)Polycyclic presentation:
3,7,19|b^a=b^2,c^a=c^747 conjugacy classes — view the full character table.
| maximal subgroup of | #11716321, #11716325, #11716327, … |
| maximal normal subgroup of | #11716321, #11716325, #11716327, … |
| maximal quotient of | C2×#87704, #11924776, C3×#87704, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(399, 3);