Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #468 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C7 · Schur multiplier (H2) = C24×C4 |
| direct product | C2×C7×#1995 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C7 | C2×D4 |
| commutator | C2×C4 | C23×C4×C7 |
| Frattini, cosocle | C22×C4 | C24×C7 |
| socle | C23×C7 | C2×C22⋊C4 |
| prime radical | C24×C7 | C2×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C7 ◁ C22×C42×C7 ◁ G | |
1206 conjugacy classes of subgroups.
| minimal degree | μ(G) = 33 |
| transitive degree | μt(G) = 448 |
A minimal faithful permutation representation — 4 generators on 33 points:
(1,2)(27,28,29,30,31,32,33);(3,4)(5,9)(6,7)(8,10)(11,12,15,18,16,19,14,24)(13,20,23,25,21,26,22,17);(4,7)(8,9)(11,13)(12,17,19,25)(14,22)(15,23)(16,21)(18,20,24,26);(3,5,6,10)(4,8,7,9)(13,21)(17,25)(20,26)(22,23)448 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 14316);