Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1909 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3×C4 · Schur multiplier (H2) = C27 |
| direct product | C2×C3×#1909 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | #386 |
| commutator | C2×C4 | C23×C3×C4 |
| Frattini, cosocle | C22×C4 | C24×C3 |
| prime radical | C22×C3×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C23×C3×C4 ◁ G | |
1686 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)(5,13)(6,17,19,10)(7,20)(8,12)(9,15)(11,22)(14,27,30,34)(16,28)(18,25,21,26)(23,29,33,31)(24,32);(3,5,7,15,8,16,22,32)(4,9,11,13,12,24,20,28)(6,14,21,31,19,30,18,29)(10,23,26,27,17,33,25,34);(3,6)(4,10)(5,14)(7,21)(8,19)(9,23)(11,26)(12,17)(13,27)(15,31)(16,30)(18,22)(20,25)(24,33)(28,34)(29,32)174 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 13995);