Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60771 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C28 |
| direct product | C3×#60773 |
| 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 ◁ C22×C3×#41 ◁ G | |
2118 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)(12,13,17,21)(14,22,23,35)(15,19,26,33)(16,28,30,20)(18,29,31,25)(24,36,37,43)(27,40,39,34)(32,41,42,38);(4,6)(5,8)(7,11)(9,10)(14,23)(18,31)(22,35)(24,37)(25,29)(32,42)(36,43)(38,41);(6,11)(8,10)(12,14,17,23)(13,18,21,31)(15,24,26,37)(16,29,30,25)(19,32,33,42)(20,35,28,22)(27,41,39,38)(34,43,40,36);(5,9)(8,10)(12,15)(13,19)(14,24)(16,27)(17,26)(18,32)(20,34)(21,33)(22,36)(23,37)(25,38)(28,40)(29,41)(30,39)(31,42)(35,43);(15,26)(19,33)(24,37)(27,39)(32,42)(34,40)(36,43)(38,41)138 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 53761);