Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61559 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C7 · Schur multiplier (H2) = C28 |
| direct product | C7×#61883 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C7 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C7 |
| prime radical | C7×#3497 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C7 ◁ C2×C4×C7×Q8 ◁ G | |
1958 conjugacy classes of subgroups.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 896 |
A minimal faithful permutation representation — 5 generators on 31 points:
(1,2,3,4,5,6,7)(9,12)(10,14)(13,15)(16,17)(18,26)(19,23)(20,24)(21,25)(22,28)(27,31)(29,30);(16,18,21,28)(17,22,25,26)(19,27,20,29)(23,30,24,31);(8,9,11,13)(10,15,14,12)(16,19,21,20)(17,23,25,24)(18,27,28,29)(22,30,26,31);(16,20,21,19)(17,24,25,23)(18,27,28,29)(22,30,26,31);(17,25)(22,26)(23,24)(30,31)427 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 54871);