Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C2 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | C24×Q8 ⋊ D3 |
| derived | C22⋊(C22⋊SL(2,𝔽3)) ⋊ C2 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | #87588 |
| commutator, cosocle, hypocenter | C22⋊(C22⋊SL(2,𝔽3)) | C2 |
| Fitting | C24×Q8 | D3 |
| socle | C25 | S4 |
| LCS | G ▷ C22⋊(C22⋊SL(2,𝔽3)) | |
| UCS | C1 ◁ C2 | |
973 conjugacy classes of subgroups; only the 23 normal subgroups are shown.
This group has 973 conjugacy classes of subgroups — too many to draw. Only its 23 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:
2,3,2^7|g^2=i,h^2=i,b^a=b^2,c^a=cd,e^a=ef,g^a=gh,h^a=hi,c^b=cd,d^b=c,e^b=ef,f^b=e,g^b=ghi,h^b=g,h^g=hi34 conjugacy classes — view the full character table.
| maximal subgroup of | #72211543, #72241468, #72241500, … |
| maximal normal subgroup of | #72211543, #72241500, #72266183, … |
| maximal quotient of | #72242977, C2×#11715844, #72267526 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1090225);