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×#2148 |
| 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 | |
1154 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);(3,4,6,10,8,13,20,12)(5,14,17,28,18,31,15,30)(7,21,19,11,23,27,32,29)(9,22,25,16,26,33,24,34);(3,5)(4,9)(6,15)(7,22)(8,18)(10,24)(11,28)(12,25)(13,26)(14,27)(16,32)(17,20)(19,34)(21,31)(23,33)(29,30);(3,6,8,20)(4,10,13,12)(5,15,18,17)(7,19,23,32)(9,24,26,25)(11,27,29,21)(14,30,31,28)(16,22,34,33)192 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 26379);