Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #576 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42×C7 · Schur multiplier (H2) = C27 |
| direct product | C7×#20348 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C7 | C22×D4 |
| commutator | C4 | C22×C42×C7 |
| Frattini, cosocle | C22×C4 | C24×C7 |
| prime radical | C22×C7×#41 | C22 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C7 ◁ C22×C7×#41 ◁ G | |
2158 conjugacy classes of subgroups.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 896 |
A minimal faithful permutation representation — 4 generators on 31 points:
(1,2,3,4,5,6,7)(8,9)(10,11)(12,13,14,15)(17,22)(19,20)(23,29)(24,25)(27,28)(30,31);(9,11)(16,17,20,24,21,25,19,22)(18,23,28,31,26,30,27,29);(16,18)(17,23)(19,27)(20,28)(21,26)(22,29)(24,31)(25,30);(18,26)(23,30)(27,28)(29,31)616 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 13336);