Finite Group DatabasebetaFinite Group Databasebeta| order | 24 |
| isoclinism | isoclinic class of Dic6 |
| automorphism | Aut(G) = D3×D4 (order 48) · Inn(G) = C2×D3 · Out(G) = C22 |
| holomorph | G ⋊ Aut(G) = #11864777 |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
| dihedral | D12 |
| unambiguous semidirect product | C4⋊D3 |
| Fitting | C3×C4 ⋊ C2 |
| Sylow | C3 ⋊ D4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C2×D3 |
| commutator, socle, cosocle | C2×C3 | C22 |
| Fitting | C3×C4 | C2 |
| hypercenter | C4 | D3 |
| hypocenter | C3 | D4 |
| LCS | G ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C2 ◁ C4 | |
16 conjugacy classes of subgroups.
| minimal degree | μ(G) = 7 |
| transitive degree | μt(G) = 12 |
A minimal faithful permutation representation — 2 generators on 7 points:
(1,2,3,4)(5,6,7);(2,4)(6,7)Polycyclic presentation:
2^3,3|b^2=c,b^a=bc,d^a=d^2| C1 | C2 | C3 | C4 | C5 | C6 | C7 | C8 | C9 | |
|---|---|---|---|---|---|---|---|---|---|
| size | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 6 | 6 |
| order | 1 | 2 | 3 | 4 | 6 | 12 | 12 | 2 | 2 |
| χ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| χ2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 |
| χ3 | 1 | 1 | 1 | -1 | 1 | -1 | -1 | 1 | -1 |
| χ4 | 1 | 1 | 1 | -1 | 1 | -1 | -1 | -1 | 1 |
| χ5 | 2 | 2 | -1 | 2 | -1 | -1 | -1 | 0 | 0 |
| χ6 | 2 | 2 | -1 | -2 | -1 | 1 | 1 | 0 | 0 |
| χ7 | 2 | -2 | 2 | 0 | -2 | 0 | 0 | 0 | 0 |
| χ8 | 2 | -2 | -1 | 0 | 1 | 0 | 0 | ||
| χ9 | 2 | -2 | -1 | 0 | 1 | 0 | 0 |
| maximal subgroup of | #204, D24, #213, … |
| maximal normal subgroup of | #204, D24, #213, … |
| maximal quotient of | #204, D24, Dic12, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(24, 6);