Finite Group DatabasebetaFinite Group Databasebeta| order | 1120 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C5×C7 · Schur multiplier (H2) = C22 |
| direct product | C5×C7×#135 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5×C7 | D4 |
| commutator, Frattini, cosocle | C4 | C23×C5×C7 |
| socle | C2×C5×C7 | C2×D4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C5×C7 ◁ C2×C4×C5×C7 ◁ G | |
124 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 560 |
A minimal faithful permutation representation — 3 generators on 28 points:
(1,2,3,4,5)(6,7,8,9,10,11,12)(13,14)(15,22)(16,20)(17,25)(18,21)(19,23)(24,27)(26,28);(13,15)(14,19)(16,23)(17,26)(18,24)(20,27)(21,28)(22,25);(13,16,18,17)(14,20,21,25)(15,23,24,26)(19,27,28,22)490 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1120, 911);