Finite Group DatabasebetaFinite Group Databasebeta| order | 392 |
| isoclinism | isoclinic class of Dic14 |
| automorphism | Aut(G) = C23×C3×AGL(1,𝔽7) (order 1008) · Inn(G) = C2×D7 · Out(G) = C22×C32 |
| homology | Abelianization (H1) = C22×C7 · Schur multiplier (H2) = C2 |
| direct product | C7×C22⋊D7 |
| unambiguous semidirect product | C22⋊(C7×D7) |
| Fitting | C22×C72 ⋊ C2 |
| Sylow | C72 ⋊ D4 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C7 | C2×D7 |
| commutator, cosocle | C2×C7 | C22×C7 |
| Frattini | C2 | C2×C7×D7 |
| Fitting | C22×C72 | C2 |
| socle | C2×C72 | C22 |
| hypercenter | C22×C7 | D7 |
| hypocenter | C7 | C7×D4 |
| LCS | G ▷ C2×C7 ▷ C7 | |
| UCS | C1 ◁ C2×C7 ◁ C22×C7 | |
47 conjugacy classes of subgroups.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 28 |
A minimal faithful permutation representation — 2 generators on 18 points:
(1,2,3,4,5,6,7)(8,9,11,13,14,12,10)(16,18);(9,10)(11,12)(13,14)(15,16)(17,18)119 conjugacy classes — view the full character table.
| maximal subgroup of | #11716108, #11716109, D7×C22⋊D7, … |
| maximal normal subgroup of | #11716108, #11716109, D7×C22⋊D7, … |
| maximal quotient of | C7×Dic7⋊C4, C7×(C2×D7)⋊C4, C7×#1141, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(392, 27);