Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C2×C42 |
No aliases for this group.
| Fitting, Sylow | #36675 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C4 | #5464 |
| commutator, cosocle, hypocenter | #2840 | C2×C3 |
| Frattini, socle | C23 | C2×C22⋊A4 |
| Fitting | #36675 | C3 |
| LCS | G ▷ #2840 | |
| UCS | C1 ◁ C4 | |
410 conjugacy classes of subgroups; only the 21 normal subgroups are shown.
This group has 410 conjugacy classes of subgroups — too many to draw. Only its 21 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|f^2=h,b^a=dgh,c^a=egh,d^a=bdfhi,e^a=cehi,g^a=gi,i^a=g,c^b=cg,d^b=dh,e^b=ei,d^c=di,e^d=egi28 conjugacy classes — view the full character table.
| maximal subgroup of | #72258191, #72263263, #72265760 |
| maximal quotient of | #72258460 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1084421);