Finite Group DatabasebetaFinite Group Databasebeta| order | 4608 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C25 |
| direct product | C3×#72209772 |
| Fitting | C3×#62817 ⋊ D3 |
| derived | #11710730 ⋊ C2×C3 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C3 | #11715850 |
| commutator, cosocle, hypocenter | #11710730 | C2×C3 |
| Frattini | C22 | C3×#87588 |
| Fitting | C3×#62817 | D3 |
| hypercenter | C22×C3 | #87588 |
| LCS | G ▷ #11710730 | |
| UCS | C1 ◁ C2×C3 ◁ C22×C3 | |
3280 conjugacy classes of subgroups; only the 42 normal subgroups are shown.
This group has 3280 conjugacy classes of subgroups — too many to draw. Only its 42 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 3 generators on 19 points:
(1,2,3)(4,5,8,11)(6,10,13,17)(7,16)(9,14)(12,19)(15,18);(6,7,12)(9,10,15)(13,14,18)(16,17,19);(4,6)(5,9)(7,12)(8,13)(10,15)(11,16)(14,18)(17,19)99 conjugacy classes — view the full character table.
| maximal subgroup of | C3×#72209832, #72231690, D3×#72209772, … |
| maximal normal subgroup of | C3×#72209832, #72231690, D3×#72209772 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,2,3)(4,5,8,11)(6,10,13,17)(7,16)(9,14)(12,19)(15,18), (6,7,12)(9,10,15)(13,14,18)(16,17,19), (4,6)(5,9)(7,12)(8,13)(10,15)(11,16)(14,18)(17,19));