Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1595 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C23 |
| direct product | C3×#8862 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C4 | #120 |
| commutator | C22×C4 | C22×C3×C4 |
| Frattini, cosocle | C2×C42 | C23×C3 |
| socle | C22×C3 | #385 |
| prime radical | C24×C3 | C4⋊C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C4 ◁ C2×C3×C42 ◁ G | |
420 conjugacy classes of subgroups.
| minimal degree | μ(G) = 43 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 3 generators on 43 points:
(1,2,3)(4,5,7,10)(6,8,11,9)(12,13,16,20,17,21,27,33)(14,18,24,30,23,29,37,41)(15,19,26,32,25,31,38,42)(22,34,36,43,35,28,39,40);(5,8)(6,11)(9,10)(12,14,16,24,17,23,27,37)(13,18,20,30,21,29,33,41)(15,22,26,36,25,35,38,39)(19,28,32,40,31,34,42,43);(12,15)(13,19)(14,22)(16,26)(17,25)(18,28)(20,32)(21,31)(23,35)(24,36)(27,38)(29,34)(30,40)(33,42)(37,39)(41,43)138 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1850);