Finite Group DatabasebetaFinite Group Databasebeta| order | 576 |
| isoclinism | isoclinic class of #1020⋊C3 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C4 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | #5936⋊C3 |
| derived | C22⋊A4 ⋊ C3×C4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | #1020⋊C3 |
| commutator, hypocenter | C22⋊A4 | C3×C4 |
| Fitting, socle | C25 | C3×D3 |
| prime radical | C2×A42 | C2 |
| cosocle | C2×C22⋊A4 | C2×C3 |
| LCS | G ▷ C22⋊A4 | |
| UCS | C1 ◁ C2 | |
103 conjugacy classes of subgroups; only the 12 normal subgroups are shown.
This group has 103 conjugacy classes of subgroups — too many to draw. Only its 12 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 12 |
Polycyclic presentation:
2,3,2,3,2^4|a^2=c,d^a=d^2,e^a=g,f^a=h,g^a=e,h^a=f,e^b=ef,f^b=e,g^b=gh,h^b=g,e^d=f,f^d=ef,g^d=gh,h^d=g28 conjugacy classes — view the full character table.
| maximal subgroup of | (#1020⋊C3)⋊C4, #11923469, #11923471, … |
| maximal normal subgroup of | (#1020⋊C3)⋊C4, #11923469, #11923471, … |
| maximal quotient of | #85561⋊C3, #85564⋊C3, #85566⋊C3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(576, 8278);