Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
| direct product | C2×#11714162 |
| Fitting | C2×#3025 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C4 | #1020⋊C2 |
| commutator, cosocle, hypocenter | #997 | C24 |
| Frattini | C23 | C23×S4 |
| Fitting | C2×#3025 | D3 |
| socle | C22 | C2×#1020⋊C2 |
| LCS | G ▷ #997 | |
| UCS | C1 ◁ C2×C4 | |
3687 conjugacy classes of subgroups; only the 107 normal subgroups are shown.
This group has 3687 conjugacy classes of subgroups — too many to draw. Only its 107 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) = 32 |
A minimal faithful permutation representation — 4 generators on 18 points:
(3,12,13,5,7,14,6,11)(4,16,17,8,10,18,9,15);(1,2)(5,6,14)(8,9,18)(11,13,12)(15,17,16);(3,4,7,10)(5,8,11,15)(6,9,13,17)(12,16,14,18);(5,11)(6,13)(8,15)(9,17)(12,14)(16,18)80 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#72217027, #72236967, #72240983, … |
| maximal normal subgroup of | C2×#72217027, #72236967, #72240983, … |
| maximal quotient of | #72246879, D3×#11714162, C4×#11714162, … |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((3,12,13,5,7,14,6,11)(4,16,17,8,10,18,9,15), (1,2)(5,6,14)(8,9,18)(11,13,12)(15,17,16), (3,4,7,10)(5,8,11,15)(6,9,13,17)(12,16,14,18), (5,11)(6,13)(8,15)(9,17)(12,14)(16,18));