Finite Group DatabasebetaFinite Group Databasebeta| order | 704 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | C22×Q8×C11 ⋊ C2 |
| Sylow | C11 ⋊ #548 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×D11 |
| commutator, socle, cosocle | C22×C11 | C24 |
| Fitting | C22×Q8×C11 | C2 |
| hypercenter | C22×Q8 | D11 |
| hypocenter | C11 | #548 |
| LCS | G ▷ C22×C11 ▷ C11 | |
| UCS | C1 ◁ C22 ◁ C22×Q8 | |
290 conjugacy classes of subgroups; only the 115 normal subgroups are shown.
This group has 290 conjugacy classes of subgroups — too many to draw. Only its 115 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 352 |
Polycyclic presentation:
2^6,11|b^2=f,c^2=f,c^a=cf,d^a=de,g^a=g^10,c^b=cf125 conjugacy classes — view the full character table.
| maximal subgroup of | #13080104, #13080111, #13080165, … |
| maximal normal subgroup of | #13080104, #13080111, #13080165, … |
| maximal quotient of | Q8×C22⋊Dic11, #13071454, #13071534, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(704, 1267);