Finite Group DatabasebetaFinite Group Databasebeta| order | 476 |
| isoclinism | isoclinic class of D7 |
| automorphism | Aut(G) = C2×C16×AGL(1,𝔽7) (order 1344) · Inn(G) = D7 · Out(G) = C2×C3×C16 |
| homology | Abelianization (H1) = C4×C17 · Schur multiplier (H2) = C1 |
| direct product | C17×Dic7 |
| unambiguous semidirect product | C7⋊(C4×C17) |
| derived, Sylow | C7 ⋊ C4×C17 |
| Hall | C7×C17 ⋊ C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C17 | D7 |
| commutator, hypocenter | C7 | C4×C17 |
| Frattini | C2 | D7×C17 |
| Fitting, socle, prime radical | C2×C7×C17 | C2 |
| cosocle | C2×C7 | C2×C17 |
| LCS | G ▷ C7 | |
| UCS | C1 ◁ C2×C17 | |
12 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 476 |
A minimal faithful permutation representation — 2 generators on 28 points:
(1,2,3,4)(6,7)(8,9)(10,11)(12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28);(5,6,8,10,11,9,7)170 conjugacy classes — view the full character table.
| maximal subgroup of | Dic7×D17, #11745751, #11745753, … |
| maximal normal subgroup of | Dic7×D17, #11745751, #11745753, … |
| maximal quotient of | C17×C7⋊C8, C2×C17×Dic7, C3×C17×Dic7, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(476, 1);