Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting | C2×#376 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center | C4 | C2×#857 |
| commutator, cosocle, hypocenter | #95⋊C3 | C23 |
| Frattini, socle | C23 | C22×S4 |
| Fitting | C2×#376 | D3 |
| hypercenter | C2×C4 | #857 |
| LCS | G ▷ #95⋊C3 | |
| UCS | C1 ◁ C4 ◁ C2×C4 | |
498 conjugacy classes of subgroups; only the 33 normal subgroups are shown.
This group has 498 conjugacy classes of subgroups — too many to draw. Only its 33 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|b^2=gh,e^2=g,f^2=ghi,c^a=cgh,d^a=d^2,e^a=efhi,f^a=fgh,g^a=gi,h^a=hi,e^d=fhi,f^d=efg,g^d=ghi,h^d=i,i^d=hi,f^e=fgh48 conjugacy classes — view the full character table.
| maximal subgroup of | #72242868, #72260665, #72260680, … |
| maximal normal subgroup of | #72278347, #72340832 |
| maximal quotient of | C4×#72985 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1086200);