Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60372 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C28 |
| direct product | C3×#60583 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C3 |
| prime radical | C3×#525 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C2×C3×#117 ◁ G | |
1998 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)(6,9)(7,8)(10,11)(12,13,17,21)(14,22,25,35)(15,19,26,33)(16,28,30,20)(18,29,32,24)(23,36,37,43)(27,40,39,34)(31,41,42,38);(5,8)(9,11);(12,14,17,25)(13,18,21,32)(15,23,26,37)(16,29,30,24)(19,31,33,42)(20,35,28,22)(27,41,39,38)(34,43,40,36);(4,6,7,10)(5,9,8,11)(12,15)(13,19)(14,23)(16,27)(17,26)(18,31)(20,34)(21,33)(22,36)(24,38)(25,37)(28,40)(29,41)(30,39)(32,42)(35,43);(15,26)(19,33)(23,37)(27,39)(31,42)(34,40)(36,43)(38,41)183 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 53571);