Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #360 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3×C4 · Schur multiplier (H2) = C26 |
| direct product | C2×C3×#2145 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C4 | #102 |
| commutator | C8 | C23×C3×C4 |
| Frattini, cosocle | C2×C8 | C24×C3 |
| socle | C22×C3 | C2×#102 |
| prime radical | C22×C3×#135 | C2 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C4 ◁ C23×C3×C4 ◁ C23×C3×C8 ◁ G | |
1170 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)(5,14,17,31)(6,10)(7,21)(8,13)(9,16,25,33)(11,19)(12,20)(15,30,18,27)(22,29,34,24)(23,26)(28,32);(3,5)(4,9)(6,15)(7,22)(8,17)(10,24)(11,27)(12,29)(13,25)(14,26)(16,32)(18,20)(19,33)(21,31)(23,34)(28,30);(3,6,8,20)(4,10,13,12)(5,15,17,18)(7,19,23,32)(9,24,25,29)(11,26,28,21)(14,30,31,27)(16,22,33,34)192 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 26375);