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×#61256 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C7 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C7 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C7 ◁ C2×C4×C7×D4 ◁ G | |
1892 conjugacy classes of subgroups.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 448 |
A minimal faithful permutation representation — 5 generators on 31 points:
(1,2,3,4,5,6,7)(8,9,12,11)(10,14,15,13)(16,17)(18,22)(19,26)(20,28)(21,24)(23,25)(27,30)(29,31);(8,10)(9,13)(11,14)(12,15)(16,18,21,20)(17,22,24,28)(19,25,27,29)(23,30,31,26);(8,11,12,9)(10,14,15,13)(16,19)(17,23)(18,25)(20,29)(21,27)(22,30)(24,31)(26,28);(10,15)(13,14)(16,18,21,20)(17,22,24,28)(19,25,27,29)(23,30,31,26);(17,24)(22,28)(23,31)(26,30)385 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 54244);