Finite Group DatabasebetaFinite Group Databasebeta| order | 272 |
| isoclinism | isoclinic class of #3633 |
| automorphism | Aut(G) = C2×AGL(1,𝔽17) (order 544) · Inn(G) = #3633 · Out(G) = C22 |
| homology | Abelianization (H1) = C16 · Schur multiplier (H2) = C1 |
No aliases for this group.
| derived, Sylow | C17 ⋊ C16 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | #3633 |
| commutator, hypocenter | C17 | C16 |
| Fitting, socle, prime radical | C2×C17 | C8 |
| cosocle | #3624 | C2 |
| LCS | G ▷ C17 | |
| UCS | C1 ◁ C2 | |
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:
(2,3,4,7,5,8,10,6)(9,13,14,11,15,16,17,12)(18,19,20,23,21,24,26,29,22,25,27,30,28,31,32,33);(1,2,6,12,10,13,17,15,8,3,9,14,16,4,11,7,5)Polycyclic presentation:
2^4,17|a^2=b,b^2=c,c^2=d,e^a=e^9,e^b=e^13,e^c=e^1620 conjugacy classes — view the full character table.
| maximal subgroup of | #10587664, #10587665, #10587666, … |
| maximal normal subgroup of | #10587664, #10587665, #10587666, … |
| maximal quotient of | #10587528, C2×#63241, C3×#63241, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(272, 28);