Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10192 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C23 |
| direct product | C3×#10207 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | #392 |
| commutator, Frattini, cosocle | C2×C42 | C23×C3 |
| prime radical | C23×C3 | #127 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C2×C3×C42 ◁ G | |
474 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 3 generators on 35 points:
(1,2,3)(4,5,10,15)(6,16,7,18)(8,14,9,13)(11,19,12,17)(21,26)(23,24)(27,30)(28,29)(31,32)(34,35);(4,6,10,7)(5,11,15,12)(8,19,9,17)(13,18,14,16)(20,21,25,29)(22,30,33,34)(23,28,24,26)(27,31,35,32);(4,7,10,6)(5,12,15,11)(8,19,9,17)(13,18,14,16)(20,22,25,33)(21,27,29,35)(23,32,24,31)(26,30,28,34)111 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 3195);