Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #58990 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C210 |
| direct product | C3×#59816 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C25 |
| commutator, Frattini, cosocle | C23 | C25×C3 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ G | |
4828 conjugacy classes of subgroups.
| minimal degree | μ(G) = 15 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 5 generators on 15 points:
(1,2,3)(5,6)(13,14);(4,5,7,6)(9,11)(13,14);(12,13,15,14);(8,9,10,11)(12,14,15,13);(4,6,7,5)192 conjugacy classes — view the full character table.
| Fitting subgroup of | D3×#59816, #72270703 |
| maximal subgroup of | D3×#59816, #72270703, C3×#7695992 |
| maximal normal subgroup of | D3×#59816, #72270703, C3×#7695992 |
| maximal quotient of | C3×#772785 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 52804);