Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61609 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C28 |
| direct product | C3×#61615 |
| 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 | |
2044 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)(15,27)(16,21)(17,30)(18,22)(20,28)(23,37)(24,33)(25,34)(26,40)(29,35)(31,36)(32,38)(39,42)(41,43);(4,6)(5,8)(7,11)(9,10)(12,14)(13,19)(15,23)(16,24)(17,26)(18,25)(20,32)(21,33)(22,34)(27,37)(28,38)(29,39)(30,40)(31,41)(35,42)(36,43);(12,15,18,29)(13,20,22,36)(14,23,25,39)(16,28,17,31)(19,32,34,43)(21,35,30,27)(24,38,26,41)(33,42,40,37);(6,11)(8,10)(12,16,18,17)(13,21,22,30)(14,24,25,26)(15,28,29,31)(19,33,34,40)(20,35,36,27)(23,38,39,41)(32,42,43,37);(5,9)(8,10)(14,25)(19,34)(23,39)(24,26)(32,43)(33,40)(37,42)(38,41)147 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 54603);