Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13077 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C24 |
| direct product | C3×#13077 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C3 | #1893 |
| commutator | C2×C8 | C22×C3×C4 |
| Frattini, cosocle | C22×C8 | C23×C3 |
| prime radical | C3×#3060 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3 ◁ C22×C3×C4 ◁ C2×C3×#131 ◁ G | |
470 conjugacy classes of subgroups; only the 146 normal subgroups are shown.
This group has 470 conjugacy classes of subgroups — too many to draw. Only its 146 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 3 generators on 35 points:
(1,2,3)(4,5)(6,16,19,31)(7,12,22,30)(8,23)(9,15)(10,28)(11,18,29,24)(13,17)(14,27)(20,21,34,25)(26,35)(32,33);(4,6,17,30,27,34,8,24,9,19,33,12,10,20,26,18)(5,11,23,21,28,7,13,31,15,29,35,25,14,22,32,16);(4,7)(5,12)(6,13)(8,25)(9,22)(10,11)(14,24)(15,30)(16,33)(17,31)(18,28)(19,32)(20,23)(21,26)(27,29)(34,35)138 conjugacy classes — view the full character table.
| Fitting subgroup of | #72270238, #72270248, D3×#13077, … |
| maximal subgroup of | C3×#495573, #72270238, #72270248, … |
| maximal normal subgroup of | C3×#495573, #72270238, #72270248, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 6065);