Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| Fitting | C22×H(5,𝔽2) ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle | C22 | C23×S4 |
| commutator, cosocle | C2×SL(2,𝔽3) | C24 |
| Fitting | C22×H(5,𝔽2) | D3 |
| hypercenter | C22×Q8 | S4 |
| hypocenter | SL(2,𝔽3) | C22×D4 |
| LCS | G ▷ C2×SL(2,𝔽3) ▷ SL(2,𝔽3) | |
| UCS | C1 ◁ C22 ◁ C22×Q8 | |
1742 conjugacy classes of subgroups; only the 137 normal subgroups are shown.
This group has 1742 conjugacy classes of subgroups — too many to draw. Only its 137 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 96 |
Polycyclic presentation:
2^5,3,2^3|b^2=i,c^2=i,g^2=i,h^2=i,d^a=de,f^a=f^2,g^a=gh,h^a=hi,c^b=ci,g^f=h,h^f=ghi,h^g=hi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72225978, #72226158, #72226217, … |
| maximal normal subgroup of | #72225978, #72226158, #72226217, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1089084);