Finite Group DatabasebetaFinite Group Databasebeta| order | 392 |
| isoclinism | isoclinic class of D7 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C7 · Schur multiplier (H2) = C23 |
| direct product | C22×C7×D7 |
| unambiguous semidirect product | C7⋊(C23×C7) |
| unambiguous semidirect product | (C2×C7)⋊(C22×C7) |
| Fitting, socle | C22×C72 ⋊ C2 |
| derived | C7 ⋊ C23×C7 |
| Sylow | C72 ⋊ C23 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22×C7 | D7 |
| commutator, cosocle, hypocenter | C7 | C23×C7 |
| Frattini | C1 | G |
| Fitting, socle | C22×C72 | C2 |
| LCS | G ▷ C7 | |
| UCS | C1 ◁ C22×C7 | |
79 conjugacy classes of subgroups.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 56 |
A minimal faithful permutation representation — 3 generators on 18 points:
(1,2)(3,4)(6,7)(8,9)(10,11)(12,13,14,15,16,17,18);(3,4)(5,6,8,10,11,9,7);(1,2)140 conjugacy classes — view the full character table.
| maximal subgroup of | (C2×D7)⋊Dic7, C7×(C2×D7)⋊C4, C2×D7×Dic7, … |
| maximal normal subgroup of | (C2×D7)⋊Dic7, C7×(C2×D7)⋊C4, C2×D7×Dic7, … |
| maximal quotient of | C2×C7×Dic14, C2×C4×C7×D7, C2×C7×D28, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(392, 42);