Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3×C4 · Schur multiplier (H2) = C27×C4 |
| direct product | C2×C3×#2922 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C42 | C23 |
| commutator | C22 | C24×C3×C4 |
| Frattini, cosocle | C2×C4 | C25×C3 |
| socle | C23×C3 | C23×C4 |
| prime radical | C22×C3×#41 | C22 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C42 ◁ G | |
2222 conjugacy classes of subgroups.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 5 generators on 29 points:
(1,2)(27,28,29);(3,4)(5,8)(6,7)(9,10)(11,12,14,17,15,18,21,24)(13,16,20,23,19,22,25,26);(4,7)(8,10)(11,13)(12,16)(14,20)(15,19)(17,23)(18,22)(21,25)(24,26);(3,5,6,9)(4,8,7,10);(13,19)(16,22)(20,25)(23,26)336 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 53107);