Finite Group DatabasebetaFinite Group Databasebeta| order | 288 |
| isoclinism | isoclinic class of #157 |
| automorphism | Aut(G) = C8×AΓL(1,𝔽9) (order 1152) · Inn(G) = #157 · Out(G) = C22×C8 |
| homology | Abelianization (H1) = C32 · Schur multiplier (H2) = C3 |
No aliases for this group.
| derived, Sylow | C32 ⋊ C32 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C8 | #157 |
| commutator, hypocenter | C32 | C32 |
| Fitting | C32×C8 | C4 |
| socle, prime radical | C2×C32 | C16 |
| cosocle | C3⋊(C3⋊C16) | C2 |
| LCS | G ▷ C32 | |
| UCS | C1 ◁ C8 | |
22 conjugacy classes of subgroups.
| minimal degree | μ(G) = 38 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 2 generators on 38 points:
(1,2)(3,5,6,4)(7,8,9,13,10,14,17,23,11,15,18,24,20,26,29,33,12,16,19,25,21,27,30,34,22,28,31,35,32,36,37,38);(1,3,6)(2,5,4)Polycyclic presentation:
2^5,3^2|a^2=b,b^2=c,c^2=d,d^2=e,f^a=fg^2,g^a=f^2g^2,f^b=f^2,g^b=g^248 conjugacy classes — view the full character table.
| maximal subgroup of | #10590011, #10590024, #10590025, … |
| maximal normal subgroup of | #10590011, #10590024, #10590025, … |
| maximal quotient of | #10589065, C2×#63540, #11720227, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(288, 188);