Finite Group DatabasebetaFinite Group Databasebeta| order | 90 |
| nilpotency class | 1 |
| isoclinism | isoclinic class of C1 |
| automorphism | Aut(G) = C4×GL(2,𝔽3) (order 192) · Inn(G) = C1 · Out(G) = C4×GL(2,𝔽3) |
| holomorph | G ⋊ Aut(G) = C2×AGL(1,𝔽5)×AGL(2,𝔽3) |
| homology | Abelianization (H1) = G · Schur multiplier (H2) = C3 |
| abelian | C2×C32×C5 |
| subgroup | G/H | |
|---|---|---|
| center | G | C1 |
| commutator, Frattini | C1 | G |
| LCS | G ▷ C1 | |
| UCS | C1 ◁ G | |
24 conjugacy classes of subgroups.
| minimal degree | μ(G) = 13 |
| transitive degree | μt(G) = 90 |
A minimal faithful permutation representation — 2 generators on 13 points:
(1,2)(3,4,5)(9,10,11,12,13);(6,7,8)90 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(90, 10);