Finite Group DatabasebetaFinite Group Databasebeta| order | 9088 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C71 ⋊ #1636 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | #14556098 |
| commutator | C42×C71 | C23 |
| Frattini | C42 | C22×D71 |
| Fitting | Q8⋊C8×C71 | C2 |
| socle | C22×C71 | #120 |
| hypercenter | Q8⋊C8 | D71 |
| hypocenter | C71 | #1636 |
| LCS | G ▷ C42×C71 ▷ C22×C71 ▷ C71 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ Q8⋊C8 | |
188 conjugacy classes of subgroups; only the 57 normal subgroups are shown.
This group has 188 conjugacy classes of subgroups — too many to draw. Only its 57 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
Polycyclic presentation:
2^7,71|a^2=f,b^2=g,c^2=f,d^2=f,e^2=g,b^a=bd,c^a=ce,d^a=df,e^a=eg,h^a=h^70,d^b=df,e^b=eg,h^b=h^70,d^c=dg,e^c=eg,h^c=h^701006 conjugacy classes — view the full character table.
| maximal subgroup of | #1636×D71, C2×#22587501, #31973213, … |
| maximal normal subgroup of | #1636×D71, C2×#22587501, #31973213, … |
| maximal quotient of | #31886318, #31886360, #31886374, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(9088, 3671);