Finite Group DatabasebetaFinite Group Databasebeta| order | 272 |
| isoclinism | isoclinic class of Dic34 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | C2×C4×C17 ⋊ C2 |
| Sylow | C17 ⋊ C22⋊C4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C2×D17 |
| commutator | C2×C17 | C2×C4 |
| Fitting | C2×C4×C17 | C2 |
| socle, cosocle | C22×C17 | C22 |
| prime radical | C22×D17 | C2 |
| hypercenter | C2×C4 | D17 |
| hypocenter | C17 | C22⋊C4 |
| LCS | G ▷ C2×C17 ▷ C17 | |
| UCS | C1 ◁ C22 ◁ C2×C4 | |
34 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)(18,19,20,21)(23,25);(2,3)(4,5)(6,7)(8,9)(10,11)(12,13)(14,15)(16,17)(22,23)(24,25)Polycyclic presentation:
2^4,17|b^2=c,b^a=bd,e^a=e^1674 conjugacy classes — view the full character table.
| maximal subgroup of | #10587558, C4×D68, #10587561, … |
| maximal normal subgroup of | #10587558, C4×D68, #10587561, … |
| maximal quotient of | #10587476, #10587477, #10587480, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(272, 14);