Finite Group DatabasebetaFinite Group Databasebeta| order | 372 |
| isoclinism | isoclinic class of D31⋊C3 |
| automorphism | Aut(G) = C2×AGL(1,𝔽31) (order 1860) · Inn(G) = D31⋊C3 · Out(G) = C2×C5 |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C2 |
| direct product | C2×D31⋊C3 |
| unambiguous semidirect product | D31⋊(C2×C3) |
| unambiguous semidirect product | (C31⋊C3)⋊C22 |
| unambiguous semidirect product | (C2×D31)⋊C3 |
| unambiguous semidirect product | (C2×C31⋊C3)⋊C2 |
| Fitting, socle | C2×C31 ⋊ C2×C3 |
| derived, Sylow | C31 ⋊ C22×C3 |
| Hall | C2×D31 ⋊ C3 |
| Hall | C31⋊C3 ⋊ C22 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | D31⋊C3 |
| commutator, cosocle, hypocenter | C31 | C22×C3 |
| Frattini | C1 | G |
| Fitting, socle | C2×C31 | C2×C3 |
| LCS | G ▷ C31 | |
| UCS | C1 ◁ C2 | |
20 conjugacy classes of subgroups.
| minimal degree | μ(G) = 33 |
| transitive degree | μt(G) = 62 |
A minimal faithful permutation representation — 2 generators on 33 points:
(4,8,9,5,6,10)(7,16,17,11,12,24)(13,26,27,19,20,32)(14,25,21,22,18,15)(23,33,30,29,28,31);(1,2)(3,4,7,13,21,9,8,14,23,30,17,25,16,20,28,27,32,33,26,12,18,24,31,29,22,6,10,15,19,11,5)22 conjugacy classes — view the full character table.
| maximal subgroup of | C4×D31⋊C3, C4⋊(D31⋊C3), (C22⋊D31)⋊C3, … |
| maximal normal subgroup of | C4×D31⋊C3, C4⋊(D31⋊C3), (C22⋊D31)⋊C3, … |
| maximal quotient of | (C31⋊C3)⋊Q8, C4×D31⋊C3, C4⋊(D31⋊C3), … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(372, 7);