Finite Group DatabasebetaFinite Group Databasebeta| order | 272 |
| isoclinism | isoclinic class of #63243 |
| automorphism | Aut(G) = D4×AGL(1,𝔽17) (order 2176) · Inn(G) = C2×#595 · Out(G) = C2×C8 |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Sylow | C17 ⋊ M16 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C2×#595 |
| commutator, socle | C2×C17 | C2×C4 |
| Fitting | C4×C17 | C4 |
| prime radical | C2×D17 | C4 |
| cosocle | Dic17 | C22 |
| hypercenter | C4 | #595 |
| hypocenter | C17 | M16 |
| LCS | G ▷ C2×C17 ▷ C17 | |
| UCS | C1 ◁ C2 ◁ C4 | |
20 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)(19,23,24,25);(2,3,4,6)(5,8,9,13)(7,12,10,11)(14,16,17,15)(18,19,21,23,22,24,20,25)Polycyclic presentation:
2^4,17|a^2=c,b^2=d,c^2=d,b^a=bd,e^a=e^13,e^c=e^1626 conjugacy classes — view the full character table.
| maximal subgroup of | #10587535, #10587536, #10587548, … |
| maximal normal subgroup of | #10587535, #10587536, #10587548, … |
| maximal quotient of | C4⋊#3624, #10587542, #10587543, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(272, 30);