Finite Group DatabasebetaFinite Group Databasebeta| order | 312 |
| isoclinism | isoclinic class of D4×C13⋊C3 |
| automorphism | Aut(G) = D4×AGL(1,𝔽13) (order 1248) · Inn(G) = C22×C13⋊C3 · Out(G) = C2×C4 |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C2 |
| direct product | D4×C13⋊C3 |
| unambiguous semidirect product | C4⋊(C2×C13⋊C3) |
| unambiguous semidirect product | (D4×C13)⋊C3 |
| Fitting, Hall | D4×C13 ⋊ C3 |
| Sylow | C13 ⋊ C3×D4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C22×C13⋊C3 |
| commutator, socle, cosocle | C2×C13 | C22×C3 |
| Fitting | D4×C13 | C3 |
| hypercenter | D4 | C13⋊C3 |
| hypocenter | C13 | C3×D4 |
| LCS | G ▷ C2×C13 ▷ C13 | |
| UCS | C1 ◁ C2 ◁ D4 | |
32 conjugacy classes of subgroups.
| minimal degree | μ(G) = 17 |
| transitive degree | μt(G) = 52 |
A minimal faithful permutation representation — 2 generators on 17 points:
(2,3,5)(4,7,10)(6,9,8)(11,13,12)(15,17);(1,2,4,3,6,10,7,11,13,5,8,12,9)(14,15)(16,17)35 conjugacy classes — view the full character table.
| maximal subgroup of | #6104⋊C3, #6105⋊C3, D8×C13⋊C3, … |
| maximal normal subgroup of | #6104⋊C3, #6105⋊C3, D8×C13⋊C3, … |
| maximal quotient of | C22⋊C4×C13⋊C3, C4⋊C4×C13⋊C3, D8×C13⋊C3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(312, 23);