Finite Group DatabasebetaFinite Group Databasebeta| order | 72 |
| isoclinism | isoclinic class of C3⋊S4 |
| automorphism | Aut(G) = ((C3⋊S4)⋊C3)⋊C2 (order 432) · Inn(G) = G · Out(G) = D3 |
| holomorph | G ⋊ Aut(G) = #72226353 |
| homology | Abelianization (H1) = C2 · Schur multiplier (H2) = C2×C3 |
| unambiguous semidirect product | C3⋊S4 |
| unambiguous semidirect product | A4⋊D3 |
| Fitting, socle | C22×C3 ⋊ D3 |
| derived | C3×A4 ⋊ C2 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | C3×A4 | C2 |
| Fitting, socle | C22×C3 | D3 |
| LCS | G ▷ C3×A4 | |
| UCS | C1 | |
30 conjugacy classes of subgroups.
| minimal degree | μ(G) = 7 |
| transitive degree | μt(G) = 12 |
A minimal faithful permutation representation — 3 generators on 7 points:
(1,2,4,3)(6,7);(5,6,7);(2,3,4)Polycyclic presentation:
2,3^2,2^2|b^a=b^2,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 | 3 | 3 | 3 | 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 | 2 | -1 | -1 | 0 | 0 |
| χ5 | 2 | -1 | 2 | -1 | -1 | 2 | -1 | 0 | 0 |
| χ6 | 2 | -1 | 2 | -1 | -1 | -1 | 2 | 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 |
| maximal subgroup of | D3×S4, C2×C3⋊S4, (C3⋊S4)⋊C3, … |
| maximal normal subgroup of | D3×S4, C2×C3⋊S4, (C3⋊S4)⋊C3, … |
| maximal quotient of | C3⋊#226, C3⋊GL(2,𝔽3), C3⋊SL(2,ℤ/4), … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(72, 43);