Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2121 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3×C4 · Schur multiplier (H2) = C25×C4 |
| direct product | C2×C3×#2122 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C2×#99 |
| commutator | C23 | C23×C3×C4 |
| Frattini, cosocle | C2×D4 | C24×C3 |
| prime radical | C2×C3×#535 | C2 |
| LCS | G ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C22×C3×#41 ◁ G | |
1406 conjugacy classes of subgroups.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 4 generators on 21 points:
(1,2)(19,20,21);(3,4)(5,9)(6,13,14,8)(7,10)(11,17)(12,16,18,15);(4,8)(5,11)(9,16)(10,13)(12,18)(15,17);(3,5,7,11)(4,9,10,17)(6,12,14,18)(8,15,13,16)138 conjugacy classes — view the full character table.
| Fitting subgroup of | C2×D3×#2122, C2×#10710957, #72254310, … |
| maximal subgroup of | C2×C3×#12755, C2×C3×#33553, C3×#498888, … |
| maximal normal subgroup of | C2×C3×#12755, C2×C3×#33553, C3×#498888, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 26333);