Finite Group DatabasebetaFinite Group Databasebeta| order | 784 |
| isoclinism | isoclinic class of C7⋊D7 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26×C7 |
| direct product | C23×C7⋊D7 |
| Fitting, socle | C23×C72 ⋊ C2 |
| derived, Sylow | C72 ⋊ C24 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C23 | C7⋊D7 |
| commutator, cosocle, hypocenter | C72 | C24 |
| Frattini | C1 | G |
| Fitting, socle | C23×C72 | C2 |
| LCS | G ▷ C72 | |
| UCS | C1 ◁ C23 | |
670 conjugacy classes of subgroups.
| transitive degree | μt(G) = 392 |
208 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#11716041, #13096407, #13096416, … |
| maximal normal subgroup of | C2×#11716041, #13096407, #13096416, … |
| maximal quotient of | C22×#87648, C22×C4×C7⋊D7, C22×C4⋊(C7⋊D7), … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(784, 171);