Finite Group DatabasebetaFinite Group Databasebeta| order | 392 |
| isoclinism | isoclinic class of D7 |
| automorphism | Aut(G) = C23×C3×AGL(1,𝔽7) (order 1008) · Inn(G) = D7 · Out(G) = C23×C32 |
| homology | Abelianization (H1) = C7×C8 · Schur multiplier (H2) = C1 |
| direct product | C7×C7⋊C8 |
| unambiguous semidirect product | C7⋊(C7×C8) |
| derived | C7 ⋊ C7×C8 |
| Sylow | C72 ⋊ C8 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C4×C7 | D7 |
| commutator, hypocenter | C7 | C7×C8 |
| Frattini | C4 | C7×D7 |
| Fitting | C4×C72 | C2 |
| socle, prime radical | C2×C72 | C4 |
| cosocle | C4×C7 | C2×C7 |
| LCS | G ▷ C7 | |
| UCS | C1 ◁ C4×C7 | |
25 conjugacy classes of subgroups.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 56 |
A minimal faithful permutation representation — 2 generators on 22 points:
(1,2,3,4,5,6,7)(9,10)(11,12)(13,14)(15,22,21,20,18,19,17,16);(8,9,11,13,14,12,10)140 conjugacy classes — view the full character table.
| maximal subgroup of | D7×C7⋊C8, #11716029, #11716030, … |
| maximal normal subgroup of | D7×C7⋊C8, #11716029, #11716030, … |
| maximal quotient of | C7×C7⋊C16, C2×C7×C7⋊C8, C3×C7×C7⋊C8, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(392, 14);