Finite Group DatabasebetaFinite Group Databasebeta| order | 728 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Hall | C7×C13 ⋊ D4 |
| Sylow | C13 ⋊ C22⋊D7 |
| Sylow | C7 ⋊ C22⋊D13 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | D7×D13 |
| commutator, Fitting, socle, cosocle | C2×C7×C13 | C22 |
| hypocenter | C7×C13 | D4 |
| LCS | G ▷ C2×C7×C13 ▷ C7×C13 | |
| UCS | C1 ◁ C2 | |
32 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 364 |
Polycyclic presentation:
2^3,7,13|b^a=bc,e^a=e^12,d^b=d^677 conjugacy classes — view the full character table.
| maximal subgroup of | #13089856, #13089862, #13089863, … |
| maximal normal subgroup of | #13089856, #13089862, #13089863, … |
| maximal quotient of | #13089759, #13089762, #13089763, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(728, 10);