Finite Group DatabasebetaFinite Group Databasebeta| order | 364 |
| isoclinism | isoclinic class of D7 |
| automorphism | Aut(G) = C2×C3×C4×AGL(1,𝔽7) (order 1008) · Inn(G) = D7 · Out(G) = C2×C32×C4 |
| homology | Abelianization (H1) = C4×C13 · Schur multiplier (H2) = C1 |
| direct product | C13×Dic7 |
| unambiguous semidirect product | C7⋊(C4×C13) |
| derived, Sylow | C7 ⋊ C4×C13 |
| Hall | C7×C13 ⋊ C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C13 | D7 |
| commutator, hypocenter | C7 | C4×C13 |
| Frattini | C2 | C13×D7 |
| Fitting, socle, prime radical | C2×C7×C13 | C2 |
| cosocle | C2×C7 | C2×C13 |
| LCS | G ▷ C7 | |
| UCS | C1 ◁ C2×C13 | |
12 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 364 |
A minimal faithful permutation representation — 2 generators on 24 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13)(14,15)(16,17)(18,19,20,21,22,23,24);(14,16,15,17)(19,24)(20,23)(21,22)130 conjugacy classes — view the full character table.
| maximal subgroup of | D13×Dic7, #10624409, #10624411, … |
| maximal normal subgroup of | D13×Dic7, #10624409, #10624411, … |
| maximal quotient of | C13×C7⋊C8, C2×C13×Dic7, C3×C13×Dic7, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(364, 1);