Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| Fitting, Sylow | #36592 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C22×#127⋊C3 |
| commutator, cosocle | C2×C42 | C23×C3 |
| Frattini, socle | C23 | C23×A4 |
| Fitting | #36592 | C3 |
| hypercenter | Q8 | #127⋊C3 |
| hypocenter | C42 | C3×#41 |
| LCS | G ▷ C2×C42 ▷ C42 | |
| UCS | C1 ◁ C2 ◁ Q8 | |
787 conjugacy classes of subgroups; only the 57 normal subgroups are shown.
This group has 787 conjugacy classes of subgroups — too many to draw. Only its 57 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 24 |
Polycyclic presentation:
3,2^8|b^2=i,c^2=gi,d^2=hi,e^2=i,f^2=i,c^a=cdfg,d^a=cefhi,e^a=eg,g^a=gh,h^a=g,c^b=cg,d^b=dh,d^c=di,f^c=fi,e^d=ei,f^e=fi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72225958, #72228058, #72228059 |
| maximal normal subgroup of | #72225958, #72228058, #72228059 |
| maximal quotient of | #72258772, #72258783 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1084394);