Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33752 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C24 |
| direct product | C3×#33754 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24×C3 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C23×C3×C4 ◁ C23×C3×C8 ◁ G | |
646 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)(6,10)(7,9)(8,11)(12,13)(14,22)(15,20)(16,19)(17,18)(21,25)(23,26)(24,27);(4,6)(5,8)(7,11)(9,10)(12,14,15,17)(13,18,20,22)(16,23,24,25)(19,21,27,26);(6,11)(8,10)(13,19)(14,17)(16,25)(18,26)(20,27)(21,22)(23,24);(5,9)(8,10)(13,20)(18,22)(19,27)(21,26)156 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 26742);