Finite Group DatabasebetaFinite Group Databasebeta| order | 2304 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C24×C3 |
| direct product | C2×#11923851 |
| Fitting, socle | C27 ⋊ C3⋊D3 |
| derived | #10596782 ⋊ C22 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | #11923851 |
| commutator, cosocle, hypocenter | #10596782 | C22 |
| Frattini | C1 | G |
| Fitting, socle | C27 | C3⋊D3 |
| LCS | G ▷ #10596782 | |
| UCS | C1 ◁ C2 | |
3990 conjugacy classes of subgroups; only the 35 normal subgroups are shown.
This group has 3990 conjugacy classes of subgroups — too many to draw. Only its 35 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 14 |
| transitive degree | μt(G) = 18 |
A minimal faithful permutation representation — 3 generators on 14 points:
(1,2)(3,4)(5,6)(7,9,8)(11,12,14);(4,6)(8,9)(12,14);(4,6,5)(8,10,9)(12,13,14)48 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#72215868, C2×#72215889, #72253634, … |
| maximal normal subgroup of | C2×#72215889, #72253634, C4×#11923851, … |
| maximal quotient of | C2×#72215868, C4×#11923851, C2×C3×#11923851 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,2)(3,4)(5,6)(7,9,8)(11,12,14), (4,6)(8,9)(12,14), (4,6,5)(8,10,9)(12,13,14));