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×#63016 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C26 |
| commutator, Frattini, cosocle | C22 | C26×C3 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ G | |
7068 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)(12,13,18,22)(14,20,15,19)(16,17,21,27)(23,26,25,24);(4,5)(6,9)(7,10)(8,11)(12,14,18,15)(13,19,22,20)(16,23,21,25)(17,24,27,26);(4,6)(5,8)(7,11)(9,10)(12,14,18,15)(13,19,22,20)(16,23,21,25)(17,24,27,26);(12,15,18,14)(13,20,22,19)(16,25,21,23)(17,26,27,24);(4,6)(5,9)(7,11)(8,10)(12,16)(13,17)(14,23)(15,25)(18,21)(19,24)(20,26)(22,27);(4,5)(6,8)(7,10)(9,11)(12,17)(13,21)(14,24)(15,26)(16,22)(18,27)(19,25)(20,23)228 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 56004);