Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2 | C2×#4626 |
| commutator, hypocenter | #95⋊C3 | C2×C4 |
| Frattini | C24 | C2×S4 |
| Fitting | #2397 | D3 |
| socle | C23 | C2×SL(2,ℤ/4) |
| prime radical | #2397⋊C3 | C2 |
| cosocle | #399⋊C3 | C22 |
| hypercenter | C4 | #4626 |
| LCS | G ▷ #95⋊C3 | |
| UCS | C1 ◁ C2 ◁ C4 | |
192 conjugacy classes of subgroups; only the 24 normal subgroups are shown.
This group has 192 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:
2^3,3,2^5|a^2=c,b^2=gh,e^2=g,f^2=ghi,b^a=bgh,d^a=d^2,e^a=ef,f^a=fgi,g^a=gi,h^a=hi,e^c=eh,f^c=fi,e^d=fh,f^d=efgh,g^d=ghi,h^d=i,i^d=hi,f^e=fgh30 conjugacy classes — view the full character table.
| maximal subgroup of | #72268835 |
| maximal normal subgroup of | #72268835 |
| maximal quotient of | #72268845 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1085756);