Finite Group DatabasebetaFinite Group Databasebeta| order | 272 |
| isoclinism | isoclinic class of Dic34 |
| automorphism | Aut(G) = C23×AGL(1,𝔽17) (order 2176) · Inn(G) = C2×D17 · Out(G) = C2×C16 |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Fitting | C8×C17 ⋊ C2 |
| Sylow | C17 ⋊ M16 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C4 | C2×D17 |
| commutator, socle | C2×C17 | C2×C4 |
| Fitting | C8×C17 | C2 |
| prime radical | C2×D17 | C4 |
| cosocle | C4×C17 | C22 |
| hypercenter | C8 | D17 |
| hypocenter | C17 | M16 |
| LCS | G ▷ C2×C17 ▷ C17 | |
| UCS | C1 ◁ C4 ◁ C8 | |
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:
(1,2,4,6,8,10,12,14,16,17,15,13,11,9,7,5,3)(18,19,20,23,21,22,24,25);(2,3)(4,5)(6,7)(8,9)(10,11)(12,13)(14,15)(16,17)(19,22)(23,25)Polycyclic presentation:
2^4,17|b^2=c,c^2=d,b^a=bd,e^a=e^1674 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#63218, #10587587, M16×D17, … |
| maximal normal subgroup of | C2×#63218, #10587587, M16×D17, … |
| maximal quotient of | #10587485, #10587486, #10587491, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(272, 5);