Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #99 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3×C4 · Schur multiplier (H2) = C211 |
| direct product | C23×C3×#99 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24×C3 | C22⋊C4 |
| commutator | C22 | C24×C3×C4 |
| Frattini, cosocle | C23 | C25×C3 |
| prime radical | C24×C3×D4 | C2 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24×C3 ◁ C26×C3 ◁ G | |
7902 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);(3,4);(5,6);(7,8)(9,13,14,11)(10,12);(7,9)(8,11)(10,14)(12,13)264 conjugacy classes — view the full character table.
| Fitting subgroup of | C23×#4741, C23×D3×#99 |
| maximal subgroup of | C23×#4741, C23×D3×#99, C23×C3×B(4,𝔽2) |
| maximal normal subgroup of | C23×#4741, C23×D3×#99, C23×C3×B(4,𝔽2) |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 53054);