Finite Group DatabasebetaFinite Group Databasebeta| order | 288 |
| isoclinism | isoclinic class of #625 |
| automorphism | Aut(G) = C2×#10596510 (order 1152) · Inn(G) = D32 · Out(G) = C2×#41 |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Sylow | C32 ⋊ M32 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C8 | D32 |
| commutator, socle | C2×C32 | C2×C8 |
| Fitting, cosocle | C32×C8 | C22 |
| prime radical | C2×C3⋊D3 | C8 |
| hypocenter | C32 | M32 |
| LCS | G ▷ C2×C32 ▷ C32 | |
| UCS | C1 ◁ C8 | |
63 conjugacy classes of subgroups.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 2 generators on 22 points:
(1,3,4,11,5,9,10,16,6,12,13,15,14,2,7,8)(18,19)(20,21,22);(1,2,4,8,5,3,10,11,6,9,13,16,14,12,7,15)(17,18,19)(21,22)Polycyclic presentation:
2^5,3^2|a^2=c,b^2=c,c^2=d,d^2=e,b^a=be,g^a=g^2,f^b=f^260 conjugacy classes — view the full character table.
| maximal subgroup of | #10590013, #10590015, #10590019, … |
| maximal normal subgroup of | #10590013, #10590015, #10590019, … |
| maximal quotient of | #10589104, #10589108, #10589133, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(288, 192);