Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #13657 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C22 |
| direct product | C3×#13658 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C3 | #2177 |
| commutator | C16 | C22×C3×C4 |
| Frattini, cosocle | C2×C16 | C23×C3 |
| prime radical | C3×#2263 | C2 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3 ◁ C2×C3×C4 ◁ C3×C4×C8 ◁ C3×#346 ◁ G | |
194 conjugacy classes of subgroups; only the 82 normal subgroups are shown.
This group has 194 conjugacy classes of subgroups — too many to draw. Only its 82 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)(5,11,13,27)(6,16)(7,19)(8,23)(9,17)(12,20,24,28)(14,26,31,30)(15,32,25,29)(18,22)(21,33)(34,35);(4,5)(6,12)(7,20)(8,24)(9,25)(10,13)(11,18)(14,23)(15,17)(16,31)(19,27)(21,26)(22,28)(29,35)(30,34)(32,33);(4,6,17,23,10,8,9,16)(5,12,15,14,13,24,25,31)(7,19,34,33,22,18,21,35)(11,26,29,20,27,30,32,28)102 conjugacy classes — view the full character table.
| maximal subgroup of | C3×AGL(1,ℤ/32), #72222504, C3×#10715810, … |
| maximal normal subgroup of | C3×AGL(1,ℤ/32), #72222504, D3×#13658, … |
| maximal quotient of | C3×#10715810, C3×#151085, C32×#13658, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 6646);