Finite Group DatabasebetaFinite Group Databasebeta| order | 272 |
| isoclinism | isoclinic class of #595 |
| automorphism | Aut(G) = D4×AGL(1,𝔽17) (order 2176) · Inn(G) = #595 · Out(G) = C4×D4 |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C2 |
No aliases for this group.
| derived, Sylow | C17 ⋊ C2×C8 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C4 | #595 |
| commutator, hypocenter | C17 | C2×C8 |
| Frattini | C2 | C2×#595 |
| Fitting | C4×C17 | C4 |
| socle | C2×C17 | C2×C4 |
| prime radical | C2×D17 | C4 |
| cosocle | Dic17 | C22 |
| LCS | G ▷ C17 | |
| UCS | C1 ◁ C4 | |
22 conjugacy classes of subgroups.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 136 |
A minimal faithful permutation representation — 2 generators on 25 points:
(1,2,5,10,6,11,15,17,13,8,14,16,12,3,7,9,4)(18,20,22,21)(19,23,25,24);(2,3,4,6)(5,8,9,13)(7,12,10,11)(14,16,17,15)(18,19,21,24,22,25,20,23)Polycyclic presentation:
2^4,17|a^2=c,b^2=d,c^2=d,e^a=e^13,e^c=e^1632 conjugacy classes — view the full character table.
| maximal subgroup of | C8×#595, #10587532, #10587547, … |
| maximal normal subgroup of | C8×#595, #10587532, #10587547, … |
| maximal quotient of | #10587529, #10587530, C4×#3624, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(272, 29);