Finite Group DatabasebetaFinite Group Databasebeta| order | 72 |
| isoclinism | isoclinic class of #157 |
| automorphism | Aut(G) = C2×AΓL(1,𝔽9) (order 288) · Inn(G) = #157 · Out(G) = C23 |
| holomorph | G ⋊ Aut(G) = #72222932 |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2×C3 |
| direct product | C2×#157 |
| Fitting, socle | C2×C32 ⋊ C4 |
| derived, Sylow | C32 ⋊ C2×C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | #157 |
| commutator, hypocenter | C32 | C2×C4 |
| Frattini | C1 | G |
| Fitting, socle | C2×C32 | C4 |
| prime radical | C2×C3⋊D3 | C2 |
| cosocle | C3⋊D3 | C22 |
| LCS | G ▷ C32 | |
| UCS | C1 ◁ C2 | |
26 conjugacy classes of subgroups.
| minimal degree | μ(G) = 8 |
| transitive degree | μt(G) = 12 |
A minimal faithful permutation representation — 2 generators on 8 points:
(3,4)(5,6,7,8);(1,2)(3,7,5)| C1 | C2 | C3 | C4 | C5 | C6 | C7 | C8 | C9 | C10 | C11 | C12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| size | 1 | 4 | 4 | 9 | 9 | 9 | 1 | 4 | 4 | 9 | 9 | 9 |
| order | 1 | 3 | 3 | 2 | 4 | 4 | 2 | 6 | 6 | 2 | 4 | 4 |
| χ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| χ2 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 |
| χ3 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | -1 | ||||
| χ4 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | -1 | ||||
| χ5 | 4 | 1 | -2 | 0 | 0 | 0 | 4 | 1 | -2 | 0 | 0 | 0 |
| χ6 | 4 | -2 | 1 | 0 | 0 | 0 | 4 | -2 | 1 | 0 | 0 | 0 |
| χ7 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 |
| χ8 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 |
| χ9 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | ||||
| χ10 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | ||||
| χ11 | 4 | 1 | -2 | 0 | 0 | 0 | -4 | -1 | 2 | 0 | 0 | 0 |
| χ12 | 4 | -2 | 1 | 0 | 0 | 0 | -4 | 2 | -1 | 0 | 0 | 0 |
Here i is the imaginary unit (i² = −1).
| derived subgroup of | #10593155, #10593156, #10593157, … |
| maximal subgroup of | D32⋊C4, #3773, #3777, … |
| maximal normal subgroup of | D32⋊C4, #3773, #3777, … |
| maximal quotient of | #3787, #3788, C4×#157, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(72, 45);