Finite Group DatabasebetaFinite Group Databasebeta| order | 1512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4×H(3,𝔽3) |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C32×C7 · Schur multiplier (H2) = C2×C32 |
| direct product | C7×D4×H(3,𝔽3) |
| unambiguous semidirect product | C4⋊(C2×C7×H(3,𝔽3)) |
| unambiguous semidirect product | (C32×D4)⋊(C3×C7) |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C3×C7 | C22×C32 |
| commutator, Frattini, cosocle | C2×C3 | C22×C32×C7 |
| LCS | G ▷ C2×C3 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C7 ◁ G | |
176 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 252 |
A minimal faithful permutation representation — 2 generators on 20 points:
(1,2,3,4,5,6,7)(8,9,10,11)(12,13,15)(14,17,19)(16,20,18);(9,11)(13,16,17)(15,19,20)385 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1512, 418);