Finite Group DatabasebetaFinite Group Databasebeta| order | 36 |
| isoclinism | isoclinic class of C3⋊D3 |
| automorphism | Aut(G) = C2×AGL(2,𝔽3) (order 864) · Inn(G) = C3⋊D3 · Out(G) = C2×S4 |
| holomorph | G ⋊ Aut(G) = #72219734 |
| homology | Abelianization (H1) = C4 · Schur multiplier (H2) = C3 |
| unambiguous semidirect product | C3⋊Dic3 |
| derived, Sylow | C32 ⋊ C4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | C3⋊D3 |
| commutator, hypocenter | C32 | C4 |
| Fitting, socle, prime radical, cosocle | C2×C32 | C2 |
| LCS | G ▷ C32 | |
| UCS | C1 ◁ C2 | |
18 conjugacy classes of subgroups.
| minimal degree | μ(G) = 10 |
| transitive degree | μt(G) = 36 |
A minimal faithful permutation representation — 3 generators on 10 points:
(1,2,3,4)(6,7)(9,10);(8,9,10);(5,6,7)Polycyclic presentation:
2^2,3^2|a^2=b,c^a=c^2,d^a=d^2| C1 | C2 | C3 | C4 | C5 | C6 | C7 | C8 | C9 | C10 | C11 | C12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| size | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 9 | 9 |
| order | 1 | 2 | 3 | 3 | 3 | 3 | 6 | 6 | 6 | 6 | 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 | -1 | -1 | ||
| χ4 | 1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | ||
| χ5 | 2 | 2 | 2 | -1 | -1 | -1 | 2 | -1 | -1 | -1 | 0 | 0 |
| χ6 | 2 | 2 | -1 | 2 | -1 | -1 | -1 | 2 | -1 | -1 | 0 | 0 |
| χ7 | 2 | 2 | -1 | -1 | 2 | -1 | -1 | -1 | 2 | -1 | 0 | 0 |
| χ8 | 2 | 2 | -1 | -1 | -1 | 2 | -1 | -1 | -1 | 2 | 0 | 0 |
| χ9 | 2 | -2 | 2 | -1 | -1 | -1 | -2 | 1 | 1 | 1 | 0 | 0 |
| χ10 | 2 | -2 | -1 | 2 | -1 | -1 | 1 | -2 | 1 | 1 | 0 | 0 |
| χ11 | 2 | -2 | -1 | -1 | 2 | -1 | 1 | 1 | -2 | 1 | 0 | 0 |
| χ12 | 2 | -2 | -1 | -1 | -1 | 2 | 1 | 1 | 1 | -2 | 0 | 0 |
Here i is the imaginary unit (i² = −1).
| derived subgroup of | #3774, #3775, #3776, … |
| maximal subgroup of | #622, D3×Dic3, #625, … |
| maximal normal subgroup of | #622, D3×Dic3, #625, … |
| maximal quotient of | C3⋊(C3⋊C8), C2×C3⋊Dic3, C3⋊Dic9, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(36, 7);