Finite Group DatabasebetaFinite Group Databasebeta| order | 312 |
| isoclinism | isoclinic class of D3 |
| automorphism | Aut(G) = C22×C3×C4×D3 (order 288) · Inn(G) = D3 · Out(G) = C22×C3×C4 |
| homology | Abelianization (H1) = C8×C13 · Schur multiplier (H2) = C1 |
| direct product | C13×C3⋊C8 |
| unambiguous semidirect product | C3⋊(C8×C13) |
| derived, Sylow | C3 ⋊ C8×C13 |
| Hall | C3×C13 ⋊ C8 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C4×C13 | D3 |
| commutator, hypocenter | C3 | C8×C13 |
| Frattini | C4 | D3×C13 |
| Fitting | C3×C4×C13 | C2 |
| socle, prime radical | C2×C3×C13 | C4 |
| cosocle | C3×C4 | C2×C13 |
| LCS | G ▷ C3 | |
| UCS | C1 ◁ C4×C13 | |
16 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 312 |
A minimal faithful permutation representation — 2 generators on 24 points:
(2,3)(4,9,6,11,7,5,10,8)(12,13,14,15,16,17,18,19,20,21,22,23,24);(1,2,3)156 conjugacy classes — view the full character table.
| maximal subgroup of | C3⋊C8×D13, #10597591, #10597592, … |
| maximal normal subgroup of | C3⋊C8×D13, #10597591, #10597592, … |
| maximal quotient of | C13×C3⋊C16, C2×C13×C3⋊C8, C13×C9⋊C8, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(312, 3);