Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30396 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C24×C4 |
| direct product | C3×#30457 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C3 |
| prime radical | C3×#1999 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ C3×C4×C22⋊C4 ◁ G | |
1118 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 4 generators on 27 points:
(1,2,3)(4,5)(6,7)(12,13,18,25,17,15,16,20)(14,21,26,19,23,27,22,24);(8,9)(10,11)(12,14,17,23)(13,19,15,24)(16,26,18,22)(20,27,25,21);(5,7)(9,11);(5,7)(12,15,17,13)(14,21,23,27)(16,20,18,25)(19,26,24,22)138 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 23445);