Finite Group DatabasebetaFinite Group Databasebeta| order | 288 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C2×C3×C4 |
No aliases for this group.
| Fitting | C32×C22⋊C4 ⋊ C2 |
| Sylow | C32 ⋊ #124 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C22×C3⋊D3 |
| commutator, socle, cosocle | C22×C32 | C23 |
| Fitting | C32×C22⋊C4 | C2 |
| prime radical | C2×#638 | C2 |
| hypercenter | C22⋊C4 | C3⋊D3 |
| hypocenter | C32 | #124 |
| LCS | G ▷ C22×C32 ▷ C32 | |
| UCS | C1 ◁ C22 ◁ C22⋊C4 | |
228 conjugacy classes of subgroups; only the 67 normal subgroups are shown.
This group has 228 conjugacy classes of subgroups — too many to draw. Only its 67 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) = 144 |
A minimal faithful permutation representation — 3 generators on 18 points:
(4,5,6)(8,10)(12,16)(15,18);(1,2,3)(7,8,9,10)(11,13,14,17)(12,15,16,18);(2,3)(5,6)(8,10)(11,12)(13,15)(14,16)(17,18)Polycyclic presentation:
2^5,3^2|b^2=d,b^a=be,c^a=cde,f^a=f^2,g^a=g^2,c^b=ce54 conjugacy classes — view the full character table.
| maximal subgroup of | #10594025, #10594034, #10594038, … |
| maximal normal subgroup of | #10594025, #10594034, #10594038, … |
| maximal quotient of | #10592345, #10592349, #10592357, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(288, 742);