Finite Group DatabasebetaFinite Group Databasebeta| order | 288 |
| automorphism | Aut(G) = D4×((C22⋊C9)⋊C2)⋊C3 (order 1728) · Inn(G) = C2×(C22⋊C9)⋊C2 · Out(G) = C22×C3 |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2 | C2×(C22⋊C9)⋊C2 |
| commutator, cosocle, hypocenter | Q8⋊C9 | C22 |
| Frattini, socle | C2×C3 | C2×S4 |
| Fitting, prime radical | C3×#41 | D3 |
| hypercenter | C4 | (C22⋊C9)⋊C2 |
| LCS | G ▷ Q8⋊C9 | |
| UCS | C1 ◁ C2 ◁ C4 | |
61 conjugacy classes of subgroups.
| minimal degree | μ(G) = 41 |
| transitive degree | μt(G) = 144 |
A minimal faithful permutation representation — 2 generators on 41 points:
(1,12,18,14,4,28,6,29,16,3,17,30)(2,19,24,11,8,31,10,25,22,7,23,32)(5,13,15,27)(9,20,21,26)(33,41,36,39,38,37,35,34,40);(1,2,6,10)(3,11,14,7)(4,9,17,21)(5,8,15,23)(12,26,29,20)(13,25,27,19)(16,24,18,22)(28,31,30,32)(34,36)(35,39)(37,38)(40,41)Polycyclic presentation:
2^2,3^2,2^3|a^2=g,b^2=g,c^3=d^2,e^2=g,f^2=g,b^a=bg,c^a=c^2d,d^a=d^2,e^a=f,f^a=e,e^c=fg,f^c=ef,f^e=fg27 conjugacy classes — view the full character table.
| maximal subgroup of | #10589505, #10589508, #10589530, … |
| maximal normal subgroup of | #10589505, #10589508, #10589530, … |
| maximal quotient of | #10589488, #10589490, C4⋊#3688, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(288, 338);