Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18132 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C25 |
| direct product | C3×#18141 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C24×C3 |
| prime radical | C3×#3459 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C23×C3×C4 ◁ G | |
1004 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)(4,5,9,13)(6,14,16,19)(7,12,8,11)(10,17,18,15)(20,21,24,27)(22,25,29,32)(23,26,30,33)(28,34,35,31);(4,6)(5,10)(7,15)(8,17)(9,16)(11,14)(12,19)(13,18)(20,22)(21,25)(23,28)(24,29)(26,31)(27,32)(30,35)(33,34);(20,23)(21,26)(22,28)(24,30)(25,31)(27,33)(29,35)(32,34);(4,7,9,8)(5,11,13,12)(6,15,16,17)(10,14,18,19)111 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 11129);