Finite Group DatabasebetaFinite Group Databasebeta| order | 272 |
| isoclinism | isoclinic class of #63228 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Sylow | C17 ⋊ Q16 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C22⋊D17 |
| commutator, cosocle | C4×C17 | C22 |
| Frattini | C4 | C2×D17 |
| Fitting | Q8×C17 | C2 |
| socle, prime radical | C2×C17 | D4 |
| hypercenter | Q8 | D17 |
| hypocenter | C17 | Q16 |
| LCS | G ▷ C4×C17 ▷ C2×C17 ▷ C17 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ Q8 | |
18 conjugacy classes of subgroups.
| minimal degree | μ(G) = 33 |
| transitive degree | μt(G) = 272 |
A minimal faithful permutation representation — 2 generators on 33 points:
(1,2,4,6,8,10,12,14,16,17,15,13,11,9,7,5,3)(18,20,22,28)(19,23,25,32)(21,30,31,27)(24,33,29,26);(2,3)(4,5)(6,7)(8,9)(10,11)(12,13)(14,15)(16,17)(18,19,22,25)(20,26,28,33)(21,29,31,24)(23,30,32,27)Polycyclic presentation:
2^4,17|a^2=d,b^2=d,c^2=d,b^a=bc,c^a=cd,e^a=e^16,c^b=cd47 conjugacy classes — view the full character table.
| maximal subgroup of | #10587601, #10587602, Q16×D17, … |
| maximal normal subgroup of | #10587601, #10587602, Q16×D17, … |
| maximal quotient of | #10587478, #10587481, #10587506, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(272, 18);