Finite Group DatabasebetaFinite Group Databasebeta| order | 6144 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C28 |
| direct product | C2×#72215886 |
| Fitting | C2×#347042 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | #72215886 |
| commutator, cosocle | #85645 | C24 |
| Frattini | C25 | C23×S4 |
| Fitting | C2×#347042 | D3 |
| socle | C23 | #11713200 |
| hypocenter | C22⋊(C22⋊A4) | C22×D4 |
| LCS | G ▷ #85645 ▷ C22⋊(C22⋊A4) | |
| UCS | C1 ◁ C2 | |
At least 11280 conjugacy classes of subgroups; only the 126 normal subgroups are shown.
This group has at least 11280 conjugacy classes of subgroups — too many to draw. Only its 126 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 — 4 generators on 18 points:
(3,4)(5,14,11,8,12,15)(6,16,7,10,13,9)(17,18);(1,2);(6,13)(10,16)(12,17)(14,18);(4,8)(9,15)(10,14)(16,18)96 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#72215896, C4×#72215886 |
| maximal normal subgroup of | C4×#72215886 |
| maximal quotient of | C4×#72215886 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((3,4)(5,14,11,8,12,15)(6,16,7,10,13,9)(17,18), (1,2), (6,13)(10,16)(12,17)(14,18), (4,8)(9,15)(10,14)(16,18));