Finite Group DatabasebetaFinite Group Databasebeta| order | 288 |
| automorphism | Aut(G) = C22×D3×S4 (order 576) · Inn(G) = D3×S4 · Out(G) = C22 |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | D3×S4 |
| commutator, cosocle, hypocenter | C3×SL(2,𝔽3) | C22 |
| Fitting | C3×Q8 | C2×D3 |
| socle | C2×C3 | C2×S4 |
| prime radical | C3⋊(#41⋊C3) | C2 |
| LCS | G ▷ C3×SL(2,𝔽3) | |
| UCS | C1 ◁ C2 | |
77 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 2 generators on 35 points:
(1,12,16,27,6,29,18,13)(2,19,22,26,10,25,24,20)(3,17,28,5,14,4,30,15)(7,23,31,9,11,8,32,21)(33,34,35);(1,2,6,10)(3,11,14,7)(4,21,16,23,15,24,17,9,18,8,5,22)(12,20,28,19,27,31,29,26,30,25,13,32)(34,35)Polycyclic presentation:
2^2,3^2,2^3|a^2=g,b^2=g,e^2=g,f^2=g,b^a=bg,d^a=d^2,c^b=c^2,e^b=f,f^b=e,e^c=fg,f^c=ef,f^e=fg21 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#64196, #10596435, #10596439, … |
| maximal normal subgroup of | C2×#64196, #10596435, #10596439, … |
| maximal quotient of | #10593194, #10593195, #10593197, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(288, 844);