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) = C2×C4×C7 · Schur multiplier (H2) = C2 |
| direct product | C4×C7×D7 |
| Fitting | C4×C72 ⋊ C2 |
| derived | C7 ⋊ C2×C4×C7 |
| Sylow | C72 ⋊ C2×C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C4×C7 | D7 |
| commutator, hypocenter | C7 | C2×C4×C7 |
| Frattini | C2 | C2×C7×D7 |
| Fitting | C4×C72 | C2 |
| socle | C2×C72 | C22 |
| prime radical | C2×C7×D7 | C2 |
| cosocle | C2×C7 | C22×C7 |
| LCS | G ▷ C7 | |
| UCS | C1 ◁ C4×C7 | |
41 conjugacy classes of subgroups.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 56 |
A minimal faithful permutation representation — 2 generators on 18 points:
(1,2,3,4,5,6,7)(9,10)(11,12)(13,14);(8,9,11,13,14,12,10)(15,16,17,18)140 conjugacy classes — view the full character table.
| maximal subgroup of | D7×C7⋊C8, #11716030, C7×C8×D7, … |
| maximal normal subgroup of | D7×C7⋊C8, #11716030, C7×C8×D7, … |
| maximal quotient of | C7×C8×D7, C7×#1131, C4×C7×Dic7, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(392, 24);