Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61629 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C28 |
| direct product | C3×#61636 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C3 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C3×#525 ◁ G | |
2074 conjugacy classes of subgroups.
| minimal degree | μ(G) = 43 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 5 generators on 43 points:
(1,2,3)(4,5)(6,10)(7,9)(8,11)(12,13)(14,19)(15,26)(16,21)(17,30)(18,22)(20,27)(23,38)(24,33)(25,34)(28,36)(29,35)(31,37)(32,41)(39,42)(40,43);(4,6)(5,8)(7,11)(9,10)(12,14,18,25)(13,19,22,34)(15,23,28,40)(16,29,17,24)(20,32,37,42)(21,35,30,33)(26,38,36,43)(27,41,31,39);(12,15)(13,20)(14,23)(16,27)(17,31)(18,28)(19,32)(21,36)(22,37)(24,39)(25,40)(26,30)(29,41)(33,38)(34,42)(35,43);(6,11)(8,10)(12,16,18,17)(13,21,22,30)(14,24,25,29)(15,27,28,31)(19,33,34,35)(20,36,37,26)(23,39,40,41)(32,38,42,43);(5,9)(8,10)(14,25)(19,34)(23,40)(24,29)(32,42)(33,35)(38,43)(39,41)156 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 54624);