Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
| direct product | C2×#11714174 |
| Fitting | C2×#3027 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#1020⋊C2 |
| commutator, cosocle, hypocenter | #997 | C24 |
| Frattini | C23 | C23×S4 |
| Fitting | C2×#3027 | D3 |
| hypercenter | C2×C4 | #1020⋊C2 |
| LCS | G ▷ #997 | |
| UCS | C1 ◁ C22 ◁ C2×C4 | |
3729 conjugacy classes of subgroups; only the 107 normal subgroups are shown.
This group has 3729 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:
(1,2)(3,11,6)(4,17,10)(7,14,16)(8,18,15);(4,8)(5,13)(6,16)(11,14);(3,4)(5,12)(6,15)(7,8)(9,13)(10,16)(11,18)(14,17);(5,11)(9,17)(12,18)(13,14)62 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#72217027, #72240989, #72240998, … |
| maximal normal subgroup of | C2×#72217027, #72240989, #72240998, … |
| maximal quotient of | #72246847, D3×#11714174, C4×#11714174, … |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,2)(3,11,6)(4,17,10)(7,14,16)(8,18,15), (4,8)(5,13)(6,16)(11,14), (3,4)(5,12)(6,15)(7,8)(9,13)(10,16)(11,18)(14,17), (5,11)(9,17)(12,18)(13,14));