Finite Group DatabasebetaFinite Group Databasebeta| order | 3072 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
| direct product | C2×#72210571 |
| Fitting | C2×#8101 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | #72210571 |
| commutator, cosocle, hypocenter | #1304⋊C3 | C23 |
| Frattini | C25 | C22×S4 |
| Fitting | C2×#8101 | D3 |
| socle | C23 | #72985 |
| LCS | G ▷ #1304⋊C3 | |
| UCS | C1 ◁ C2 | |
3955 conjugacy classes of subgroups; only the 39 normal subgroups are shown.
This group has 3955 conjugacy classes of subgroups — too many to draw. Only its 39 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 — 3 generators on 18 points:
(3,15,13,4,10,9)(5,8,6,12,7,14)(11,17,16);(6,11,10,16)(7,8)(12,13)(14,18,15,17);(1,2)(3,4)(5,9)(6,14)(7,8)(10,15)(11,17)(12,13)(16,18)58 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#72210665 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((3,15,13,4,10,9)(5,8,6,12,7,14)(11,17,16), (6,11,10,16)(7,8)(12,13)(14,18,15,17), (1,2)(3,4)(5,9)(6,14)(7,8)(10,15)(11,17)(12,13)(16,18));