Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8346 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C25 |
| direct product | C3×#8346 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | #379 |
| commutator | C22×C4 | C22×C3×C4 |
| Frattini, cosocle | C23×C4 | C23×C3 |
| prime radical | C3×#3490 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C23×C3×C4 ◁ G | |
1052 conjugacy classes of subgroups.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 3 generators on 19 points:
(1,2,3)(5,9,11,18)(6,13)(10,17,12,14)(15,19);(4,5)(6,14)(7,10)(8,12)(9,15)(11,16)(13,18)(17,19);(6,13)(9,17)(14,18)(15,19)138 conjugacy classes — view the full character table.
| Fitting subgroup of | D3×#8346, #72252292 |
| maximal subgroup of | D3×#8346, #72252292, C3×#346446, … |
| maximal normal subgroup of | D3×#8346, #72252292, C3×#346446 |
| maximal quotient of | C3×#10686697 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1334);