Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
| direct product | C2×#11713242 |
| Fitting | C2×C4×#120 ⋊ D3 |
4154 conjugacy classes of subgroups; only the 116 normal subgroups are shown.
This group has 4154 conjugacy classes of subgroups — too many to draw. Only its 116 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) = 48 |
A minimal faithful permutation representation — 4 generators on 18 points:
(1,2)(4,10,8)(5,9,6);(11,13,14,17)(12,15,16,18);(6,9)(7,10)(13,17)(15,18);(3,4)(5,8)(6,7)(9,10)(11,12,14,16)(13,15,17,18)76 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#72220618, C2×#72220641, C2×#72220624 |
| maximal normal subgroup of | C2×#72220618, C2×#72220641, C2×#72220624 |
| maximal quotient of | C2×#72245595 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,2)(4,10,8)(5,9,6), (11,13,14,17)(12,15,16,18), (6,9)(7,10)(13,17)(15,18), (3,4)(5,8)(6,7)(9,10)(11,12,14,16)(13,15,17,18));