Finite Group DatabasebetaFinite Group Databasebeta| order | 288 |
| isoclinism | isoclinic class of D32 |
| automorphism | Aut(G) = C2×#10596510 (order 1152) · Inn(G) = D32 · Out(G) = C2×#41 |
| homology | Abelianization (H1) = C2×C16 · Schur multiplier (H2) = C2 |
No aliases for this group.
| derived, Sylow | C32 ⋊ C2×C16 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C8 | D32 |
| commutator, hypocenter | C32 | C2×C16 |
| Fitting, cosocle | C32×C8 | C22 |
| socle | C2×C32 | C2×C8 |
| prime radical | C2×C3⋊D3 | C8 |
| LCS | G ▷ C32 | |
| UCS | C1 ◁ C8 | |
67 conjugacy classes of subgroups.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 2 generators on 22 points:
(1,3,4,10,5,11,12,16,6,2,7,8,13,9,14,15)(18,19)(20,21,22);(1,2,4,8,5,9,12,15,6,3,7,10,13,11,14,16)(17,18,19)(21,22)Polycyclic presentation:
2^5,3^2|a^2=c,b^2=c,c^2=d,d^2=e,g^a=g^2,f^b=f^272 conjugacy classes — view the full character table.
| maximal subgroup of | D32⋊C16, #10590014, #10590016, … |
| maximal normal subgroup of | D32⋊C16, #10590014, #10590016, … |
| maximal quotient of | #10589067, #10589069, #10589102, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(288, 190);