Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2204 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C26 |
| direct product | C2×C3×#2204 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | B(4,𝔽2) |
| commutator, Frattini, cosocle | C2×Q8 | C24×C3 |
| LCS | G ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C23×C3 ◁ C22×C3×Q8 ◁ G | |
1200 conjugacy classes of subgroups; only the 222 normal subgroups are shown.
This group has 1200 conjugacy classes of subgroups — too many to draw. Only its 222 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 4 generators on 21 points:
(1,2)(19,20,21);(3,4)(5,11)(6,9)(7,15)(8,12)(10,14)(13,18)(16,17);(4,9)(5,7)(6,14)(10,12)(11,17)(15,18);(3,5)(6,13)(7,8)(9,10)(11,15)(14,16)120 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 26553);