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.
| derived | Q8⋊C9 ⋊ C22 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C2×(C22⋊C9)⋊C2 |
| commutator, cosocle, hypocenter | Q8⋊C9 | C22 |
| Frattini, socle | C2×C3 | C2×S4 |
| Fitting | C3×#41 | D3 |
| hypercenter | C4 | (C22⋊C9)⋊C2 |
| LCS | G ▷ Q8⋊C9 | |
| UCS | C1 ◁ C2 ◁ C4 | |
71 conjugacy classes of subgroups.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 72 |
A minimal faithful permutation representation — 2 generators on 25 points:
(1,7,12,6,3,15,5,13,10,2,11,16)(4,8,9,14)(17,25,20,23,22,21,19,18,24);(2,6)(3,4)(7,14)(8,13)(9,11)(10,12)(18,20)(19,23)(21,22)(24,25)Polycyclic presentation:
2^2,3^2,2^3|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 | #10589508, #10589509, #10589531, … |
| maximal normal subgroup of | #10589508, #10589509, #10589531, … |
| maximal quotient of | #10589493, C4⋊#3689, #10589496, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(288, 340);