Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting, Sylow | #36575 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | #85645 |
| commutator, cosocle, hypocenter | C24×Q8 | C2×C3 |
| Frattini, socle | C23 | C2×C22⋊A4 |
| Fitting | #36575 | C3 |
| LCS | G ▷ C24×Q8 | |
| UCS | C1 ◁ C2 | |
949 conjugacy classes of subgroups; only the 24 normal subgroups are shown.
This group has 949 conjugacy classes of subgroups — too many to draw. Only its 24 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 48 |
Polycyclic presentation:
3,2^8|b^2=i,c^2=i,d^2=i,e^2=i,f^2=i,c^a=cdei,d^a=cf,e^a=fi,f^a=efi,g^a=gh,h^a=g,c^b=cg,d^b=dh,d^c=di,f^c=fi,e^d=ei,f^e=fi34 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1084383);