Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12503 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C2×C8 |
| direct product | C3×#12517 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | #493 |
| commutator, Frattini, cosocle | C4×C8 | C23×C3 |
| prime radical | C3×#1712 | C2 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C3×C42 ◁ C3×C82 ◁ G | |
354 conjugacy classes of subgroups; only the 126 normal subgroups are shown.
This group has 354 conjugacy classes of subgroups — too many to draw. Only its 126 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 3 generators on 27 points:
(1,2,3)(5,8)(6,9)(10,11)(13,17)(14,21)(15,22)(18,23)(19,27)(20,26)(24,25);(12,13,18,27,15,19,14,17,16,20,21,25,22,24,23,26);(4,5,9,11,7,10,6,8)210 conjugacy classes — view the full character table.
| maximal subgroup of | C3×#491469, D3×#12517, #72225924, … |
| maximal normal subgroup of | C3×#491469, D3×#12517, #72225924, … |
| maximal quotient of | C3×#123419, C3×#123411 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 5505);