Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #63041 |
| automorphism | ? |
| homology | Abelianization (H1) = C26×C3 · Schur multiplier (H2) = C213 |
| direct product | C3×#63052 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C26 |
| commutator, Frattini, cosocle | C22 | C26×C3 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ G | |
7124 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)(6,7)(8,9)(10,11)(12,13)(14,22)(15,17)(16,20)(18,19)(21,25)(23,27)(24,26);(5,9)(6,10)(13,19)(15,25)(20,22)(23,26);(4,6,8,10)(5,7,9,11)(12,14,18,16)(13,20,19,22)(15,23,25,26)(17,24,21,27);(4,7)(5,10)(6,9)(8,11)(12,15)(13,21)(14,23)(16,26)(17,19)(18,25)(20,27)(22,24);(12,16,18,14)(13,20,19,22)(15,26,25,23)(17,24,21,27);(12,17,18,21)(13,15,19,25)(14,24,16,27)(20,23,22,26)210 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 56040);