Finite Group DatabasebetaFinite Group Databasebeta| order | 1080 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4×H(3,𝔽3) |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C32×C5 · Schur multiplier (H2) = C32 |
| direct product | C5×Q8×H(3,𝔽3) |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C3×C5 | C22×C32 |
| commutator, Frattini, cosocle | C2×C3 | C22×C32×C5 |
| prime radical | C2×C5×H(3,𝔽3) | C22 |
| LCS | G ▷ C2×C3 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C5 ◁ G | |
132 conjugacy classes of subgroups.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 360 |
A minimal faithful permutation representation — 2 generators on 22 points:
(1,2,3,4,5)(6,7,8,9)(10,13,12,11)(14,15,17)(16,19,21)(18,22,20);(6,10,8,12)(7,11,9,13)(15,18,19)(17,21,22)275 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1080, 250);