Finite Group DatabasebetaFinite Group Databasebeta| order | 72 |
| isoclinism | isoclinic class of (C22⋊C9)⋊C2 |
| automorphism | Aut(G) = ((C22⋊C9)⋊C2)⋊C3 (order 216) · Inn(G) = G · Out(G) = C3 |
| holomorph | G ⋊ Aut(G) = #72220207 |
| homology | Abelianization (H1) = C2 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | (C22⋊C9)⋊C2 |
| derived | C22⋊C9 ⋊ C2 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | C22⋊C9 | C2 |
| Frattini | C3 | S4 |
| Fitting, socle | C22×C3 | D3 |
| LCS | G ▷ C22⋊C9 | |
| UCS | C1 | |
18 conjugacy classes of subgroups.
| minimal degree | μ(G) = 13 |
| transitive degree | μt(G) = 18 |
A minimal faithful permutation representation — 2 generators on 13 points:
(1,2,4,3)(6,8)(7,11)(9,10)(12,13);(2,3,4)(5,13,8,11,10,9,7,6,12)Polycyclic presentation:
2,3^2,2^2|b^3=c^2,b^a=b^2c,c^a=c^2,d^a=e,e^a=d,d^b=e,e^b=de| C1 | C2 | C3 | C4 | C5 | C6 | C7 | C8 | C9 | |
|---|---|---|---|---|---|---|---|---|---|
| size | 1 | 2 | 3 | 6 | 8 | 8 | 8 | 18 | 18 |
| order | 1 | 3 | 2 | 6 | 9 | 9 | 9 | 2 | 4 |
| χ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| χ2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 |
| χ3 | 2 | 2 | 2 | 2 | -1 | -1 | -1 | 0 | 0 |
| χ4 | 2 | -1 | 2 | -1 | ζ+ζ8 | ζ2+ζ7 | ζ4+ζ5 | 0 | 0 |
| χ5 | 2 | -1 | 2 | -1 | ζ2+ζ7 | ζ4+ζ5 | ζ+ζ8 | 0 | 0 |
| χ6 | 2 | -1 | 2 | -1 | ζ4+ζ5 | ζ+ζ8 | ζ2+ζ7 | 0 | 0 |
| χ7 | 3 | 3 | -1 | -1 | 0 | 0 | 0 | 1 | -1 |
| χ8 | 3 | 3 | -1 | -1 | 0 | 0 | 0 | -1 | 1 |
| χ9 | 6 | -3 | -2 | 1 | 0 | 0 | 0 | 0 | 0 |
Here ζ is a primitive root of unity of order 9 (ζ9 = 1).
| maximal subgroup of | C2×(C22⋊C9)⋊C2, ((C22⋊C9)⋊C2)⋊C3, C3×(C22⋊C9)⋊C2, … |
| maximal normal subgroup of | C2×(C22⋊C9)⋊C2, ((C22⋊C9)⋊C2)⋊C3, C3×(C22⋊C9)⋊C2, … |
| maximal quotient of | #3688, #3689, (C22⋊C9)⋊C4, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(72, 15);