Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | #857⋊C2 |
| commutator | C2×C42⋊C3 | C2×C4 |
| Frattini, socle | C24 | C2×S4 |
| Fitting | C2×#384 | D3 |
| prime radical, cosocle | C22×C42⋊C3 | C22 |
| hypocenter | C42⋊C3 | C4⋊C4 |
| LCS | G ▷ C2×C42⋊C3 ▷ C42⋊C3 | |
| UCS | C1 ◁ C22 | |
346 conjugacy classes of subgroups; only the 28 normal subgroups are shown.
This group has 346 conjugacy classes of subgroups — too many to draw. Only its 28 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:
2^4,3,2^4|a^2=d,b^2=c,f^2=h,g^2=i,b^a=bd,e^a=e^2,f^a=fgi,g^a=gi,h^a=hi,f^b=fh,g^b=gi,f^e=gh,g^e=fg,h^e=i,i^e=hi44 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#11711337, #72214621, #72214668, … |
| maximal normal subgroup of | C2×#11711337, #72218177, #72218188, … |
| maximal quotient of | #72214561, C2×#11711337, #72214621, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1085718);