Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #29634 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C24×C4 |
| direct product | C3×#29637 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C2×#120 |
| commutator, Frattini, cosocle | C24 | C24×C3 |
| prime radical | C3×#3449 | C2 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C23×C3×C4 ◁ G | |
1358 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)(4,5)(6,9)(7,8)(10,11)(12,13)(14,18)(15,19)(16,20)(17,21)(22,27)(23,25)(24,26);(5,8)(9,11)(12,14)(13,18)(15,22)(16,23)(17,24)(19,27)(20,25)(21,26);(4,6)(5,9)(7,10)(8,11)(12,15,17,16)(13,19,21,20)(14,22,24,23)(18,25,26,27);(6,10)(9,11)(12,16,17,15)(13,20,21,19)(14,23,24,22)(18,25,26,27)111 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 22625);