Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C24 |
| direct product | C2×#11714281 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | #85559 |
| commutator, cosocle, hypocenter | #2679⋊C3 | C22 |
| Frattini | C23 | C2×#1020 |
| Fitting | C2×#2679 | D3 |
| socle | C24 | #1020 |
| prime radical | C2×#2679⋊C3 | C2 |
| LCS | G ▷ #2679⋊C3 | |
| UCS | C1 ◁ C22 | |
1361 conjugacy classes of subgroups; only the 23 normal subgroups are shown.
This group has 1361 conjugacy classes of subgroups — too many to draw. Only its 23 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) = 24 |
A minimal faithful permutation representation — 2 generators on 14 points:
(1,2)(3,5,14,6,10,9)(4,7,13,8,12,11);(3,4)(5,14,10,9)(6,8)(7,13)(11,12)46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#72240889, C2×#72240888, C2×#72240893, … |
| maximal normal subgroup of | C2×#72240889, C2×#72240888, C2×#72240893, … |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,2)(3,5,14,6,10,9)(4,7,13,8,12,11), (3,4)(5,14,10,9)(6,8)(7,13)(11,12));