Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | #3486 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #85493 |
| commutator, cosocle | C22×SL(2,𝔽3) | C23 |
| Frattini | C23 | C22×S4 |
| Fitting | #3486 | D3 |
| hypercenter | #120 | S4 |
| hypocenter | SL(2,𝔽3) | #120 |
| LCS | G ▷ C22×SL(2,𝔽3) ▷ SL(2,𝔽3) | |
| UCS | C1 ◁ C2 ◁ C23 ◁ #120 | |
777 conjugacy classes of subgroups; only the 49 normal subgroups are shown.
This group has 777 conjugacy classes of subgroups — too many to draw. Only its 49 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
Polycyclic presentation:
2^5,3,2^3|g^2=i,h^2=i,c^a=ce,d^a=di,f^a=f^2,g^a=gh,h^a=hi,c^b=cd,e^b=ei,g^f=h,h^f=ghi,h^g=hi44 conjugacy classes — view the full character table.
| maximal subgroup of | #72241523 |
| maximal normal subgroup of | #72241523 |
| maximal quotient of | #72241493 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1087568);