Finite Group DatabasebetaFinite Group Databasebeta| order | 272 |
| isoclinism | isoclinic class of #595 |
| automorphism | Aut(G) = C4×AGL(1,𝔽17) (order 1088) · Inn(G) = #595 · Out(G) = C42 |
| homology | Abelianization (H1) = C16 · Schur multiplier (H2) = C1 |
No aliases for this group.
| derived, Sylow | C17 ⋊ C16 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C4 | #595 |
| commutator, hypocenter | C17 | C16 |
| Fitting | C4×C17 | C4 |
| socle, prime radical | C2×C17 | C8 |
| cosocle | #3622 | C2 |
| LCS | G ▷ C17 | |
| UCS | C1 ◁ C4 | |
10 conjugacy classes of subgroups.
| minimal degree | μ(G) = 33 |
| transitive degree | μt(G) = 272 |
A minimal faithful permutation representation — 2 generators on 33 points:
(2,3,4,6)(5,8,9,13)(7,12,10,11)(14,16,17,15)(18,19,20,23,21,24,26,29,22,25,27,30,28,31,32,33);(1,2,5,10,6,11,15,17,13,8,14,16,12,3,7,9,4)Polycyclic presentation:
2^4,17|a^2=b,b^2=c,c^2=d,e^a=e^13,e^b=e^1632 conjugacy classes — view the full character table.
| maximal subgroup of | #10587528, #10587529, #10587530, … |
| maximal normal subgroup of | #10587528, #10587529, #10587530, … |
| maximal quotient of | #10587467, C2×#63216, C3×#63216, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(272, 3);