Finite Group DatabasebetaFinite Group Databasebeta| order | 312 |
| isoclinism | isoclinic class of #261 |
| automorphism | Aut(G) = C22×AGL(1,𝔽13) (order 624) · Inn(G) = #261 · Out(G) = C22×C3 |
| homology | Abelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C2 |
| direct product | C2×C3×#261 |
| Fitting, socle | C2×C3×C13 ⋊ C4 |
| derived, Sylow | C13 ⋊ C2×C3×C4 |
| Hall | C3×C13 ⋊ C2×C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C3 | #261 |
| commutator, hypocenter | C13 | C2×C3×C4 |
| Frattini | C1 | G |
| Fitting, socle | C2×C3×C13 | C4 |
| prime radical | C2×C3×D13 | C2 |
| cosocle | D13 | C22×C3 |
| LCS | G ▷ C13 | |
| UCS | C1 ◁ C2×C3 | |
32 conjugacy classes of subgroups.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 78 |
A minimal faithful permutation representation — 2 generators on 18 points:
(3,4,5)(7,8,9,11)(10,13,14,18)(12,16,17,15);(1,2)(6,7,10,13,17,11,15,16,8,12,18,14,9)42 conjugacy classes — view the full character table.
| maximal subgroup of | Dic3×#261, (C2×D3)⋊#261, Dic3⋊#261, … |
| maximal normal subgroup of | Dic3×#261, (C2×D3)⋊#261, Dic3⋊#261, … |
| maximal quotient of | C3×D13⋊C8, C3×#6118, C3×C4×#261, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(312, 52);