Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2192 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C26 |
| direct product | C2×C3×#2193 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | #447 |
| commutator, Frattini, cosocle | C2×C8 | C24×C3 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C2×C3×#131 ◁ G | |
1098 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);(4,10)(5,14)(6,17)(7,8)(11,25)(12,29)(13,24)(15,31)(16,21)(18,22)(19,20)(23,26)(27,34)(28,32);(3,4)(5,11)(6,18)(7,12)(8,22)(9,13)(10,17)(14,26)(15,32)(16,28)(19,24)(20,29)(21,27)(23,30)(25,31)(33,34);(3,5,9,16)(4,11,13,28)(6,15,20,31)(7,21,8,14)(10,23,24,34)(12,27,22,26)(17,30,19,33)(18,32,29,25)138 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 26524);