Finite Group DatabasebetaFinite Group Databasebeta| order | 576 |
| isoclinism | isoclinic class of #10588306 |
| automorphism | ? |
| homology | Abelianization (H1) = C4 · Schur multiplier (H2) = C4 |
No aliases for this group.
| derived | C42⋊C9 ⋊ C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, hypocenter | C42⋊C9 | C4 |
| Frattini | C23×C3 | S4 |
| Fitting, prime radical | C3×#127 | D3 |
| socle | C22×C3 | SL(2,ℤ/4) |
| cosocle | #127⋊C9 | C2 |
| LCS | G ▷ C42⋊C9 | |
| UCS | C1 | |
63 conjugacy classes of subgroups.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 72 |
Polycyclic presentation:
2^2,3^2,2^4|a^2=b,c^3=d^2,e^2=gh,f^2=g,c^a=c^2d,d^a=d^2,e^a=f,f^a=egh,g^a=gh,e^b=egh,f^b=fg,e^c=fh,f^c=efgh,g^c=h,h^c=gh21 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#10588306, #10588306⋊C2, C3×#10588306, … |
| maximal normal subgroup of | C2×#10588306, #10588306⋊C2, C3×#10588306, … |
| maximal quotient of | #11919927, #11919928, #11919931, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(576, 188);