Finite Group DatabasebetaFinite Group Databasebeta| order | 3456 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C23 |
| direct product | C2×#10593110⋊C3 |
| derived | #63703 ⋊ C22×C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | #11724050 |
| commutator, cosocle, hypocenter | #63703 | C22×C3 |
| Frattini | C2×C3 | C2×C3×#1020 |
| Fitting | C2×C32×H(5,𝔽2) | D3 |
| socle | C22×C3 | C3×#1020 |
| LCS | G ▷ #63703 | |
| UCS | C1 ◁ C22 | |
1395 conjugacy classes of subgroups; only the 49 normal subgroups are shown.
This group has 1395 conjugacy classes of subgroups — too many to draw. Only its 49 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) = 144 |
A minimal faithful permutation representation — 2 generators on 19 points:
(1,2)(4,8,10)(5,9,6)(11,15,16,13,19,18,17,12,14);(3,6,9,5,7,4,10,8)(12,14)(13,17)(15,18)(16,19)112 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#72219864, #72253113, C2×#72252942 |
| maximal normal subgroup of | #72253113 |
| maximal quotient of | C2×#72252942 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,2)(4,8,10)(5,9,6)(11,15,16,13,19,18,17,12,14), (3,6,9,5,7,4,10,8)(12,14)(13,17)(15,18)(16,19));