Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1504 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3×C4 · Schur multiplier (H2) = C24×C4 |
| direct product | C2×C3×#1521 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C2×C22⋊C4 |
| commutator | C2×C4 | C23×C3×C4 |
| Frattini, cosocle | C22×C4 | C24×C3 |
| prime radical | C2×C3×#121 | C22 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ C22×C3×C42 ◁ G | |
1074 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)(6,19)(10,25,26,31)(12,24,13,20)(15,30,27,22)(17,33)(18,29)(21,32)(28,34);(3,4)(5,15)(6,20)(7,12)(8,13)(9,21)(10,18)(11,16)(14,26)(17,30)(19,23)(22,33)(24,32)(25,34)(27,29)(28,31);(3,5,16,29)(4,10,11,26)(6,17,32,33)(7,18,8,14)(9,22,23,30)(12,15,13,27)(19,34,21,28)(20,25,24,31)192 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 10554);