Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| isoclinism | isoclinic class of #11710845 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting, Sylow | #63040 ⋊ C3 |
| derived | H(3,𝔽4) ⋊ C22×C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | H(3,𝔽4) | C22×C3 |
| Frattini, socle | C22 | C22×C22⋊A4 |
| Fitting | #63040 | C3 |
| LCS | G ▷ H(3,𝔽4) | |
| UCS | C1 | |
1488 conjugacy classes of subgroups; only the 41 normal subgroups are shown.
This group has 1488 conjugacy classes of subgroups — too many to draw. Only its 41 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 24 |
Polycyclic presentation:
3,2^8|b^2=i,c^2=i,d^2=i,e^2=h,f^2=i,g^2=i,b^a=degh,c^a=ceg,d^a=cefi,e^a=ceh,f^a=bcei,g^a=ehi,h^a=hi,i^a=h,c^b=ch,d^b=di,e^b=eh,d^c=dhi,g^f=gi36 conjugacy classes — view the full character table.
| maximal subgroup of | #72235037, #72235439, #72260994, … |
| maximal normal subgroup of | #72235037, #72235439, #72262057, … |
| maximal quotient of | #72235026, #72283789, #72340893, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1085226);