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×#61614 |
| 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×#525 ◁ G | |
2120 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 5 generators on 27 points:
(1,2,3)(4,5)(6,10)(7,9)(8,11)(12,13,16,20)(14,21,23,26)(15,19,17,24)(18,25,27,22);(4,6)(5,8)(7,11)(9,10)(12,13)(14,18)(15,24)(16,20)(17,19)(21,22)(23,27)(25,26);(12,14)(13,18)(15,22)(16,23)(17,25)(19,26)(20,27)(21,24);(6,11)(8,10)(12,15,16,17)(13,19,20,24)(14,22,23,25)(18,26,27,21);(5,9)(8,10)(13,20)(18,27)(19,24)(21,26)147 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 54602);