Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #63011 |
| automorphism | ? |
| homology | Abelianization (H1) = C26×C3 · Schur multiplier (H2) = C213 |
| direct product | C3×#63021 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C26 |
| commutator, Frattini, cosocle | C22 | C26×C3 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ G | |
7272 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 6 generators on 27 points:
(1,2,3)(4,5)(6,9)(7,10)(8,11)(12,13)(14,22)(15,17)(16,19)(18,21)(20,26)(23,27)(24,25);(4,5)(6,9)(7,10)(8,11)(12,14,18,16)(13,19,21,22)(15,23,26,25)(17,24,20,27);(4,6)(5,8)(7,11)(9,10)(12,14,18,16)(13,19,21,22)(15,23,26,25)(17,24,20,27);(12,15)(13,20)(14,23)(16,25)(17,21)(18,26)(19,27)(22,24);(4,6)(5,9)(7,11)(8,10)(12,16,18,14)(13,19,21,22)(15,25,26,23)(17,24,20,27);(4,5)(6,8)(7,10)(9,11)(12,17,18,20)(13,15,21,26)(14,24,16,27)(19,23,22,25)228 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 56009);