Finite Group DatabasebetaFinite Group Databasebeta| order | 364 |
| isoclinism | isoclinic class of D13 |
| automorphism | Aut(G) = C22×C3×AGL(1,𝔽13) (order 1872) · Inn(G) = D13 · Out(G) = C23×C32 |
| homology | Abelianization (H1) = C22×C7 · Schur multiplier (H2) = C2 |
| direct product | C2×C7×D13 |
| unambiguous semidirect product | C13⋊(C22×C7) |
| unambiguous semidirect product | (C2×C13)⋊(C2×C7) |
| Fitting, socle | C2×C7×C13 ⋊ C2 |
| derived, Sylow | C13 ⋊ C22×C7 |
| Hall | C7×C13 ⋊ C22 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C7 | D13 |
| commutator, cosocle, hypocenter | C13 | C22×C7 |
| Frattini | C1 | G |
| Fitting, socle | C2×C7×C13 | C2 |
| LCS | G ▷ C13 | |
| UCS | C1 ◁ C2×C7 | |
20 conjugacy classes of subgroups.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 182 |
A minimal faithful permutation representation — 2 generators on 22 points:
(1,2)(3,4,5,6,7,8,9)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22);(1,2)(10,11,13,15,17,19,21,22,20,18,16,14,12)112 conjugacy classes — view the full character table.
| maximal subgroup of | D13×Dic7, #10624410, #10624411, … |
| maximal normal subgroup of | D13×Dic7, #10624410, #10624411, … |
| maximal quotient of | C7×Dic26, C4×C7×D13, C7×D52, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(364, 8);