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×#62159 |
| 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×#529 ◁ G | |
1712 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 5 generators on 35 points:
(1,2,3)(4,5,8,11)(6,12,13,9)(7,15,16,10)(14,17,19,18)(20,21,24,30)(22,31,33,26)(23,28,25,29)(27,34,35,32);(4,6)(5,9)(7,14)(8,13)(10,17)(11,12)(15,18)(16,19)(20,22)(21,26)(23,34)(24,33)(25,32)(27,28)(29,35)(30,31);(20,22)(21,27)(23,32)(24,33)(25,34)(26,28)(29,31)(30,35);(6,13)(9,12)(14,19)(17,18)(20,23,24,25)(21,28,30,29)(22,32,33,34)(26,35,31,27);(4,7,8,16)(5,10,11,15)(6,14,13,19)(9,17,12,18)(20,23,24,25)(21,29,30,28)(22,32,33,34)(26,27,31,35)147 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 55147);