Finite Group DatabasebetaFinite Group Databasebeta| order | 288 |
| isoclinism | isoclinic class of D3×S4 |
| automorphism | Aut(G) = C22×D3×S4 (order 576) · Inn(G) = D3×S4 · Out(G) = C22 |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C22 |
| unambiguous semidirect product | (C3⋊S4)⋊C4 |
| derived | C3×A4 ⋊ C2×C4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | D3×S4 |
| commutator, hypocenter | C3×A4 | C2×C4 |
| Fitting, socle | C23×C3 | C2×D3 |
| prime radical | C2×C3⋊S4 | C2 |
| cosocle | C2×C3×A4 | C22 |
| LCS | G ▷ C3×A4 | |
| UCS | C1 ◁ C2 | |
122 conjugacy classes of subgroups; only the 21 normal subgroups are shown.
This group has 122 conjugacy classes of subgroups — too many to draw. Only its 21 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 11 |
| transitive degree | μt(G) = 36 |
A minimal faithful permutation representation — 2 generators on 11 points:
(1,3,4,2)(5,6,8,7)(9,10,11);(1,2,4,3)(6,7,8)(10,11)Polycyclic presentation:
2^3,3^2,2^2|a^2=c,b^2=c,e^a=e^2,d^b=d^2,f^b=g,g^b=f,f^d=g,g^d=fg30 conjugacy classes — view the full character table.
| maximal subgroup of | #10596421, (C3⋊S4)⋊Q8, #10596428, … |
| maximal normal subgroup of | #10596421, (C3⋊S4)⋊Q8, #10596428, … |
| maximal quotient of | (C3⋊S4)⋊C8, #10593176, #10593193, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(288, 854);