Finite Group DatabasebetaFinite Group Databasebeta| order | 108 |
| isoclinism | isoclinic class of ((C3⋊D3)⋊C3)⋊C2 |
| automorphism | Aut(G) = #1088⋊C2 (order 216) · Inn(G) = G · Out(G) = C2 |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | ((C3⋊D3)⋊C3)⋊C2 |
| unambiguous semidirect product | #272⋊C2 |
| Fitting, derived, Sylow | H(3,𝔽3) ⋊ C22 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, Fitting, cosocle, hypocenter | H(3,𝔽3) | C22 |
| Frattini, socle | C3 | D32 |
| LCS | G ▷ H(3,𝔽3) | |
| UCS | C1 | |
39 conjugacy classes of subgroups.
| minimal degree | μ(G) = 9 |
| transitive degree | μt(G) = 9 |
A minimal faithful permutation representation — 2 generators on 9 points:
(2,8,6,4,5,9)(3,7);(1,2,4)(3,5,8,7,6,9)Polycyclic presentation:
2^2,3^3|d^a=d^2,e^a=e^2,c^b=c^2,e^b=e^2,d^c=de| C1 | C2 | C3 | C4 | C5 | C6 | C7 | C8 | C9 | C10 | C11 | |
|---|---|---|---|---|---|---|---|---|---|---|---|
| size | 1 | 2 | 6 | 6 | 9 | 9 | 9 | 12 | 18 | 18 | 18 |
| order | 1 | 3 | 3 | 3 | 2 | 2 | 2 | 3 | 6 | 6 | 6 |
| χ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| χ2 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 |
| χ3 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 |
| χ4 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 |
| χ5 | 2 | 2 | 2 | -1 | 0 | 0 | 2 | -1 | 0 | 0 | -1 |
| χ6 | 2 | 2 | 2 | -1 | 0 | 0 | -2 | -1 | 0 | 0 | 1 |
| χ7 | 2 | 2 | -1 | 2 | 0 | 2 | 0 | -1 | 0 | -1 | 0 |
| χ8 | 2 | 2 | -1 | 2 | 0 | -2 | 0 | -1 | 0 | 1 | 0 |
| χ9 | 4 | 4 | -2 | -2 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| χ10 | 6 | -3 | 0 | 0 | 2 | 0 | 0 | 0 | -1 | 0 | 0 |
| χ11 | 6 | -3 | 0 | 0 | -2 | 0 | 0 | 0 | 1 | 0 | 0 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(108, 17);