Finite Group DatabasebetaFinite Group Databasebeta| order | 408 |
| isoclinism | isoclinic class of D17 |
| automorphism | Aut(G) = C23×AGL(1,𝔽17) (order 2176) · Inn(G) = D17 · Out(G) = C23×C8 |
| homology | Abelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C2 |
| direct product | C3×C4×D17 |
| Fitting | C3×C4×C17 ⋊ C2 |
| derived, Sylow | C17 ⋊ C2×C3×C4 |
| Hall | C3×C17 ⋊ C2×C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3×C4 | D17 |
| commutator, hypocenter | C17 | C2×C3×C4 |
| Frattini | C2 | C2×C3×D17 |
| Fitting | C3×C4×C17 | C2 |
| socle | C2×C3×C17 | C22 |
| prime radical | C2×C3×D17 | C2 |
| cosocle | C2×C17 | C22×C3 |
| LCS | G ▷ C17 | |
| UCS | C1 ◁ C3×C4 | |
32 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 204 |
A minimal faithful permutation representation — 2 generators on 24 points:
(2,3)(4,5)(6,7)(8,9)(10,11)(12,13)(14,15)(16,17)(22,23,24);(1,2,4,6,8,10,12,14,16,17,15,13,11,9,7,5,3)(18,21,20,19)120 conjugacy classes — view the full character table.
| maximal subgroup of | C3⋊C8×D17, #11717740, C3×C8×D17, … |
| maximal normal subgroup of | C3⋊C8×D17, #11717740, C3×C8×D17, … |
| maximal quotient of | C3×C8×D17, C3×#63218, C3×C4×Dic17, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(408, 16);