Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1913 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3×C4 · Schur multiplier (H2) = C27 |
| direct product | C2×C3×#1915 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | #386 |
| commutator | C23 | C23×C3×C4 |
| Frattini, cosocle | C22×C4 | C24×C3 |
| prime radical | C22×C3×#41 | C22 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C23×C3×C4 ◁ G | |
1670 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);(3,4,7,12,8,11,13,26)(5,9,16,23,15,24,20,27)(6,17,19,28,18,10,21,25)(14,22,30,33,29,32,31,34);(3,5,13,16,8,15,7,20)(4,9,12,27,11,24,26,23)(6,14,19,31,18,29,21,30)(10,22,28,33,17,32,25,34);(3,6)(4,10)(5,14)(7,21)(8,18)(9,22)(11,17)(12,28)(13,19)(15,29)(16,31)(20,30)(23,34)(24,32)(25,26)(27,33)174 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 14004);