Finite Group DatabasebetaFinite Group Databasebeta| order | 784 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22×C7 |
No aliases for this group.
| Fitting | C72×Q8 ⋊ C2 |
| Sylow | C72 ⋊ #41 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C22×C7⋊D7 |
| commutator, socle, cosocle | C2×C72 | C23 |
| Fitting | C72×Q8 | C2 |
| hypercenter | Q8 | C7⋊D7 |
| hypocenter | C72 | #41 |
| LCS | G ▷ C2×C72 ▷ C72 | |
| UCS | C1 ◁ C2 ◁ Q8 | |
200 conjugacy classes of subgroups.
| transitive degree | μt(G) = 392 |
Polycyclic presentation:
2^4,7^2|b^2=d,c^2=d,c^a=cd,e^a=e^6,f^a=f^6,c^b=cd130 conjugacy classes — view the full character table.
| maximal subgroup of | #13096222, #13096379, #13096383, … |
| maximal normal subgroup of | #13096222, #13096379, #13096383, … |
| maximal quotient of | #13096531, #13096533, #13096534, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(784, 153);