Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C29 |
No aliases for this group.
| Fitting | C7×#1781 ⋊ C2 |
| Sylow | C7 ⋊ #32898 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | C2×#6445 |
| commutator, cosocle | C24×C7 | C24 |
| Frattini | C24 | C23×D7 |
| Fitting | C7×#1781 | C2 |
| socle | C22×C7 | C2×#120 |
| hypercenter | #1781 | D7 |
| hypocenter | C7 | #32898 |
| LCS | G ▷ C24×C7 ▷ C2×C7 ▷ C7 | |
| UCS | C1 ◁ C22 ◁ C25 ◁ #1781 | |
4360 conjugacy classes of subgroups; only the 255 normal subgroups are shown.
This group has 4360 conjugacy classes of subgroups — too many to draw. Only its 255 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 224 |
Polycyclic presentation:
2^8,7|b^a=be,c^a=cf,d^a=dg,i^a=i^6,f^b=fh,e^c=eh,i^c=i^6145 conjugacy classes — view the full character table.
| maximal subgroup of | D7×#32898, #72277956 |
| maximal normal subgroup of | D7×#32898, #72277956 |
| maximal quotient of | #72260017, #72260036 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 318341);