Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #453 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3×C4 · Schur multiplier (H2) = C28 |
| direct product | C2×C3×#1792 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | #120 |
| commutator | C2×C4 | C23×C3×C4 |
| Frattini, cosocle | C22×C4 | C24×C3 |
| prime radical | C22×C3×H(5,𝔽2) | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ C23×C3×C4 ◁ G | |
4114 conjugacy classes of subgroups.
| minimal degree | μ(G) = 17 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 17 points:
(1,2)(15,16,17);(3,4,5,6)(8,11)(9,13)(12,14);(7,8)(9,14)(10,12)(11,13);(9,13)(11,14)192 conjugacy classes — view the full character table.
| maximal subgroup of | C2×C3×#13343, AGL(1,𝔽7)×#1792 |
| maximal normal subgroup of | C2×C3×#13343 |
| maximal quotient of | C2×C3×#8453, AGL(1,𝔽7)×#1792 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 13601);