Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1665 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C24 |
| direct product | C2×C3×#1678 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | #121 |
| commutator, Frattini, cosocle | C42 | C24×C3 |
| prime radical | C2×C3×#127 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ C2×C3×C42 ◁ G | |
722 conjugacy classes of subgroups.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 4 generators on 29 points:
(1,2)(27,28,29);(4,7)(5,9)(8,10)(11,12)(13,19)(14,16)(15,23)(17,21)(18,22)(20,26)(24,25);(3,4,5,7,6,8,9,10)(12,16)(17,25)(18,23)(19,26);(11,13,14,20)(12,17,16,25)(15,24,22,21)(18,26,23,19)174 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 12107);