Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D4×AGL(1,ℤ/8) |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C7 · Schur multiplier (H2) = C28 |
| direct product | C7×#61470 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C7 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C7 |
| prime radical | C7×#3482 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C7 ◁ C4×C7×#41 ◁ G | |
1812 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)(13,15)(16,17,21,24)(18,22,20,29)(19,26,28,30)(23,27,31,25);(8,9)(10,13)(11,12)(14,15)(16,18,21,20)(17,22,24,29)(19,27,28,25)(23,26,31,30);(16,19,21,28)(17,23,24,31)(18,25,20,27)(22,30,29,26);(9,12)(13,15)(17,24)(22,29)(23,31)(26,30);(8,10,11,14)(9,13,12,15)(17,24)(22,29)(23,31)(26,30)385 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 54458);