Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
No aliases for this group.
| Fitting | H(7,𝔽2) ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle | C2 | C24×S4 |
| commutator, cosocle, hypocenter | SL(2,𝔽3) | C25 |
| Fitting | H(7,𝔽2) | D3 |
| hypercenter | 2−1+4 | S4 |
| LCS | G ▷ SL(2,𝔽3) | |
| UCS | C1 ◁ C2 ◁ 2−1+4 | |
5274 conjugacy classes of subgroups; only the 509 normal subgroups are shown.
This group has 5274 conjugacy classes of subgroups — too many to draw. Only its 509 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
Polycyclic presentation:
2^5,3,2^3|c^2=i,e^2=i,g^2=i,h^2=i,e^a=ei,f^a=f^2,g^a=gh,h^a=hi,d^b=di,e^c=ei,g^f=h,h^f=ghi,h^g=hi83 conjugacy classes — view the full character table.
| maximal subgroup of | #72228369 |
| maximal normal subgroup of | #72228369 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1090177);