Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #353 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C16 · Schur multiplier (H2) = C2 |
| direct product | C3×#7337 |
| subgroup | G/H | |
|---|---|---|
| center | C3×C8 | #99 |
| commutator | C2×C4 | C2×C3×C16 |
| Frattini, cosocle | C2×M32 | C22×C3 |
| socle | C2×C3 | #1314 |
| prime radical | C2×C3×#41 | C8 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C3×C8 ◁ C2×C3×C8 ◁ C2×C3×M32 ◁ G | |
154 conjugacy classes of subgroups; only the 54 normal subgroups are shown.
This group has 154 conjugacy classes of subgroups — too many to draw. Only its 54 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 67 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 2 generators on 67 points:
(1,2,3)(4,5,8,14,9,16,28,41,10,13,27,37,30,38,54,61,11,15,29,40,31,42,55,63,7,22,24,39,25,49,51,62)(6,17,19,43,20,44,47,64,18,12,32,33,45,35,56,58,21,36,26,52,48,60,53,66,23,34,46,57,50,59,65,67);(4,6)(5,12)(7,23)(8,26)(9,20)(10,18)(11,21)(13,36)(14,33)(15,34)(16,35)(17,22)(19,29)(24,32)(25,50)(27,46)(28,53)(30,45)(31,48)(37,52)(38,60)(39,43)(40,57)(41,58)(42,59)(44,49)(47,55)(51,56)(54,65)(61,66)(62,64)(63,67)156 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 325);