Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C24 |
| direct product | C2×#8541⋊C3 |
| Fitting, Sylow | C2×#8541 ⋊ C3 |
| derived | C26 ⋊ C2×C3×C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | #8541⋊C3 |
| commutator, hypocenter | C26 | C2×C3×C4 |
| Frattini | C25 | C22×A4 |
| Fitting | C2×#8541 | C3 |
| socle | C23 | #379⋊C3 |
| prime radical | C2×#85645 | C2 |
| cosocle | C22×#120 | C22×C3 |
| LCS | G ▷ C26 | |
| UCS | C1 ◁ C2 | |
3974 conjugacy classes of subgroups; only the 31 normal subgroups are shown.
This group has 3974 conjugacy classes of subgroups — too many to draw. Only its 31 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 24 |
A minimal faithful permutation representation — 2 generators on 18 points:
(1,2)(6,13,7)(9,10,16)(11,12,17)(14,15,18);(3,16,5,8)(4,17,18,13)(6,9,12,15)(7,10,11,14)48 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#72215851, C2×#72215879, C2×#72215885, … |
| maximal normal subgroup of | C2×#72215851, C2×#72215879, C2×#72215885, … |
| maximal quotient of | C4×#8541⋊C3, C2×#72216921, C2×C3×#8541⋊C3 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,2)(6,13,7)(9,10,16)(11,12,17)(14,15,18), (3,16,5,8)(4,17,18,13)(6,9,12,15)(7,10,11,14));