Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19621 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C23×C4 |
| direct product | C3×#19625 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C2×#127 |
| commutator, Frattini, cosocle | C42 | C24×C3 |
| prime radical | C3×#3205 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C2×C3×C42 ◁ G | |
836 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 27 points:
(1,2,3)(5,8)(6,10)(9,11)(12,13)(14,20)(15,17)(16,24)(18,22)(19,23)(21,26)(25,27);(4,5,6,8,7,9,10,11)(13,17)(18,25)(19,24)(20,26);(12,14,16,22,15,21,23,27)(13,18,19,20,17,25,24,26);(5,9)(8,11)156 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 12613);