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×#60499 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C3 |
| prime radical | C2×C3×2−1+4 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C3×C4×#41 ◁ G | |
1970 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)(4,5,7,9)(6,10,11,8)(12,13,16,18)(14,22,24,35)(15,20,26,32)(17,29,28,21)(19,30,31,25)(23,36,37,43)(27,40,39,34)(33,41,42,38);(4,6,7,11)(5,8,9,10)(13,18)(19,31)(20,32)(21,29)(22,35)(33,42)(34,40)(36,43);(12,14,16,24)(13,19,18,31)(15,23,26,37)(17,30,28,25)(20,33,32,42)(21,35,29,22)(27,41,39,38)(34,43,40,36);(12,15)(13,20)(14,23)(16,26)(17,27)(18,32)(19,33)(21,34)(22,36)(24,37)(25,38)(28,39)(29,40)(30,41)(31,42)(35,43);(15,26)(20,32)(23,37)(27,39)(33,42)(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, 53487);