Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2403 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C210×C4 |
| direct product | C2×C3×#2433 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24×C3 | C24 |
| commutator, Frattini, cosocle | C23 | C25×C3 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24×C3 ◁ G | |
9422 conjugacy classes of subgroups.
| minimal degree | μ(G) = 17 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 5 generators on 17 points:
(1,2)(15,16,17);(4,5)(8,9);(3,4,6,5)(7,8,10,9)(12,14);(7,8,10,9)(11,12)(13,14);(3,5,6,4)(7,9,10,8)228 conjugacy classes — view the full character table.
| Fitting subgroup of | C2×#10990483, C2×D3×#2433, C2×#11632790, … |
| maximal subgroup of | C2×#10990483, C2×D3×#2433, C2×C3×D4×#120, … |
| maximal normal subgroup of | C2×#10990483, C2×D3×#2433, C2×C3×D4×#120, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 27661);