Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19621 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C7 · Schur multiplier (H2) = C24×C4 |
| direct product | C7×#19631 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C7 | C2×#127 |
| commutator, Frattini, cosocle | C42 | C24×C7 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C7 ◁ C2×C42×C7 ◁ G | |
922 conjugacy classes of subgroups.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 448 |
A minimal faithful permutation representation — 4 generators on 31 points:
(1,2,3,4,5,6,7)(9,12)(10,13)(14,15)(16,17)(18,24)(19,27)(20,22)(21,26)(23,28)(25,30)(29,31);(16,18,19,26,20,25,28,31)(17,21,23,24,22,29,27,30);(8,9,13,15,11,14,10,12)(17,22)(21,29)(23,27)(24,30);(18,25)(21,29)(24,30)(26,31)364 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 12619);