Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C24 |
| direct product | C3×#19387 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C2×#121 |
| commutator, Frattini, cosocle | C42 | C24×C3 |
| prime radical | C22×C3×D4 | D4 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C2×C3×C42 ◁ G | |
654 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 35 points:
(1,2,3)(4,5,8,11)(6,12,15,19)(7,16,13,10)(9,17,18,14)(20,21)(22,25)(23,26)(24,28)(27,31)(29,33)(30,32)(34,35);(4,6,13,17,8,15,7,14)(5,9,16,19,11,18,10,12)(20,22)(21,25)(23,30)(24,29)(26,32)(27,34)(28,33)(31,35);(21,26)(23,31)(25,32)(27,28)(30,35)(33,34);(22,29)(25,33)(30,35)(32,34)147 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 12375);