Finite Group DatabasebetaFinite Group Databasebeta| order | 288 |
| isoclinism | isoclinic class of #63745 |
| automorphism | Aut(G) = #11880560 (order 1152) · Inn(G) = #3777 · Out(G) = C23 |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C3 |
No aliases for this group.
| Sylow | C32 ⋊ #108 |
38 conjugacy classes of subgroups.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 2 generators on 25 points:
(1,4,5,6)(3,8,7,9)(10,12,19,20,14,21,13,22)(11,15,23,18,17,24,16,25);(2,4,5,9)(3,7,8,6)(10,11,13,16,14,17,19,23)(12,18,20,24,21,25,22,15)Polycyclic presentation:
2^5,3^2|a^2=c,b^2=cd,c^2=e,b^a=bc,f^a=fg^2,g^a=f^2g^2,c^b=ce,d^b=de,f^b=g,g^b=f^2,f^c=f^2,g^c=g^218 conjugacy classes — view the full character table.
| maximal subgroup of | #10593167, #10593168, #10593169, … |
| maximal normal subgroup of | #10593167, #10593168, #10593169, … |
| maximal quotient of | #10590088, #10590089, #10590097, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(288, 396);