Finite Group DatabasebetaFinite Group Databasebeta| order | 312 |
| isoclinism | isoclinic class of D13⋊C3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C23 |
| direct product | C22×D13⋊C3 |
| unambiguous semidirect product | D13⋊(C22×C3) |
| unambiguous semidirect product | (C13⋊C3)⋊C23 |
| unambiguous semidirect product | (C2×C13⋊C3)⋊C22 |
| Fitting, socle | C22×C13 ⋊ C2×C3 |
| derived, Sylow | C13 ⋊ C23×C3 |
| Hall | C22×D13 ⋊ C3 |
| Hall | C13⋊C3 ⋊ C23 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | D13⋊C3 |
| commutator, cosocle, hypocenter | C13 | C23×C3 |
| Frattini | C1 | G |
| Fitting, socle | C22×C13 | C2×C3 |
| LCS | G ▷ C13 | |
| UCS | C1 ◁ C22 | |
64 conjugacy classes of subgroups.
| minimal degree | μ(G) = 17 |
| transitive degree | μt(G) = 52 |
A minimal faithful permutation representation — 3 generators on 17 points:
(6,10,11,7,8,12)(9,16,15,13,14,17);(3,4)(5,6,9,8,12,15,14,16,17,11,10,13,7);(1,2)32 conjugacy classes — view the full character table.
| maximal subgroup of | #6103⋊C3, (C22⋊#261)⋊C3, C2×C4×D13⋊C3, … |
| maximal normal subgroup of | #6103⋊C3, (C22⋊#261)⋊C3, C2×C4×D13⋊C3, … |
| maximal quotient of | C2×(C13⋊C3)⋊Q8, C2×C4×D13⋊C3, C2×C4⋊(D13⋊C3), … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(312, 49);