Finite Group DatabasebetaFinite Group Databasebeta| order | 18176 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | C71×#2262 ⋊ C2 |
| Sylow | C71 ⋊ #13742 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C2×#17985834 |
| commutator, cosocle | C32×C71 | C23 |
| Frattini | C32 | C22×D71 |
| Fitting | C71×#2262 | C2 |
| socle | C2×C71 | C2×D32 |
| hypercenter | #2262 | D71 |
| hypocenter | C71 | #13742 |
| LCS | G ▷ C32×C71 ▷ C16×C71 ▷ C8×C71 ▷ C4×C71 ▷ C2×C71 ▷ C71 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ C2×C16 ◁ C2×C32 ◁ #2262 | |
134 conjugacy classes of subgroups; only the 47 normal subgroups are shown.
This group has 134 conjugacy classes of subgroups — too many to draw. Only its 47 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 4544 |
Polycyclic presentation:
2^8,71|d^2=ef,e^2=fg,f^2=gh,g^2=h,b^a=bd,c^a=ch,d^a=de,e^a=ef,f^a=fg,g^a=gh,d^b=de,e^b=ef,f^b=fg,g^b=gh,i^b=i^701376 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(18176, 90625);