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×#63015 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C26 |
| commutator, Frattini, cosocle | C22 | C26×C3 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ G | |
7166 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)(12,13,18,23)(14,21,16,19)(15,17,22,20)(24,27,26,25);(4,5)(6,9)(7,10)(8,11)(12,14,18,16)(13,19,23,21)(15,24,22,26)(17,25,20,27);(4,6)(5,8)(7,11)(9,10)(12,15)(13,20)(14,24)(16,26)(17,23)(18,22)(19,27)(21,25);(12,16,18,14)(13,21,23,19)(15,26,22,24)(17,27,20,25);(4,6)(5,9)(7,11)(8,10)(12,15)(13,17)(14,24)(16,26)(18,22)(19,25)(20,23)(21,27);(4,5)(6,8)(7,10)(9,11)(12,17)(13,22)(14,25)(15,23)(16,27)(18,20)(19,26)(21,24)228 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 56003);