Finite Group DatabasebetaFinite Group Databasebeta| order | 72 |
| isoclinism | isoclinic class of #634 |
| automorphism | Aut(G) = C22×AGL(2,𝔽3) (order 1728) · Inn(G) = C2×C3⋊D3 · Out(G) = GL(2,𝔽3) |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2×C3 |
No aliases for this group.
| Fitting | C22×C32 ⋊ C2 |
| Sylow | C32 ⋊ D4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C2×C3⋊D3 |
| commutator, socle, cosocle | C2×C32 | C22 |
| Fitting | C22×C32 | C2 |
| hypercenter | C22 | C3⋊D3 |
| hypocenter | C32 | D4 |
| LCS | G ▷ C2×C32 ▷ C32 | |
| UCS | C1 ◁ C2 ◁ C22 | |
48 conjugacy classes of subgroups.
| minimal degree | μ(G) = 10 |
| transitive degree | μt(G) = 36 |
A minimal faithful permutation representation — 3 generators on 10 points:
(2,4)(8,9,10);(1,2)(3,4)(6,7)(9,10);(5,6,7)Polycyclic presentation:
2^3,3^2|b^a=bc,d^a=d^2,e^a=e^221 conjugacy classes — view the full character table.
| maximal subgroup of | #3804, D3×#67, #3828, … |
| maximal normal subgroup of | #3804, D3×#67, #3828, … |
| maximal quotient of | #3750, #3752, #3753, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(72, 35);