Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1510 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3×C4 · Schur multiplier (H2) = C24×C4 |
| direct product | C2×C3×#1515 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C2×C22⋊C4 |
| commutator | C2×C4 | C23×C3×C4 |
| Frattini, cosocle | C22×C4 | C24×C3 |
| prime radical | C24×C3 | C2×D4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ C22×C3×C42 ◁ G | |
1022 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 37 points:
(1,2)(35,36,37);(4,9,11,23)(5,14)(6,18)(10,25,26,34)(12,24,13,19)(16,31)(17,30)(20,21)(22,27,33,28)(29,32);(3,4,7,12)(5,10,17,27)(6,19,18,23)(8,13,15,11)(9,21,24,20)(14,28,30,26)(16,25,32,33)(22,31,34,29);(3,5,15,30)(4,10,11,26)(6,16,21,31)(7,17,8,14)(9,22,23,33)(12,27,13,28)(18,32,20,29)(19,25,24,34)156 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 10545);