Finite Group DatabasebetaFinite Group Databasebeta| order | 704 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | C11×#135 ⋊ C2 |
| Sylow | C11 ⋊ #577 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C2×D4×D11 |
| commutator, cosocle | C4×C11 | C24 |
| Frattini | C4 | C23×D11 |
| Fitting | C11×#135 | C2 |
| socle | C2×C11 | C22×D4 |
| hypercenter | #135 | D11 |
| hypocenter | C11 | #577 |
| LCS | G ▷ C4×C11 ▷ C2×C11 ▷ C11 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ #135 | |
258 conjugacy classes of subgroups; only the 99 normal subgroups are shown.
This group has 258 conjugacy classes of subgroups — too many to draw. Only its 99 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 176 |
Polycyclic presentation:
2^6,11|d^2=f,e^2=f,b^a=bf,g^a=g^10,c^b=cf,d^b=df,d^c=de,e^c=ef,e^d=ef92 conjugacy classes — view the full character table.
| maximal subgroup of | #13080380, #13080383, #13080387, … |
| maximal normal subgroup of | #13080380, #13080383, #13080387, … |
| maximal quotient of | #13077138, #13077145, #13077208, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(704, 1220);