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×#1912 |
| 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 | |
1694 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);(3,4,8,12,9,13,7,23)(5,14,18,25,17,29,16,10)(6,19,21,27,22,11,20,26)(15,28,32,24,31,34,30,33);(3,5)(4,10,13,25)(6,15)(7,16)(8,18)(9,17)(11,24,19,33)(12,14,23,29)(20,30)(21,32)(22,31)(26,34,27,28);(3,6,9,22)(4,11,13,19)(5,15,17,31)(7,21,8,20)(10,24,25,33)(12,27,23,26)(14,28,29,34)(16,32,18,30)174 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 13999);