Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60460 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C27×C4 |
| direct product | C3×#60627 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C3 |
| prime radical | C3×#3499 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C3×C4×#41 ◁ G | |
1998 conjugacy classes of subgroups.
| minimal degree | μ(G) = 43 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 5 generators on 43 points:
(1,2,3)(4,5)(6,9)(7,8)(10,11)(12,13,19,25)(14,21,28,34)(15,20,27,35)(16,22,30,37)(17,24,18,23)(26,36,40,43)(29,33,41,42)(31,39,32,38);(5,8)(9,11)(12,14,17,15,19,28,18,27)(13,20,23,21,25,35,24,34)(16,26,31,29,30,40,32,41)(22,33,38,36,37,42,39,43);(4,6,7,10)(5,9,8,11)(12,15,17,28,19,27,18,14)(13,21,23,35,25,34,24,20)(16,29,31,40,30,41,32,26)(22,36,38,42,37,43,39,33);(12,16)(13,22)(14,26)(15,29)(17,31)(18,32)(19,30)(20,33)(21,36)(23,38)(24,39)(25,37)(27,41)(28,40)(34,43)(35,42);(16,30)(22,37)(26,40)(29,41)(31,32)(33,42)(36,43)(38,39)174 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 53615);