Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| isoclinism | isoclinic class of #85547 |
| automorphism | ? |
| homology | Abelianization (H1) = C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| derived | C22⋊(C42⋊C3) ⋊ C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | #85547 |
| commutator, hypocenter | C22⋊(C42⋊C3) | C4 |
| Frattini | C23 | #1020 |
| Fitting | C23×C42 | D3 |
| socle | C25 | S4 |
| prime radical, cosocle | C2×C22⋊(C42⋊C3) | C2 |
| LCS | G ▷ C22⋊(C42⋊C3) | |
| UCS | C1 ◁ C2 | |
523 conjugacy classes of subgroups; only the 19 normal subgroups are shown.
This group has 523 conjugacy classes of subgroups — too many to draw. Only its 19 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 24 |
Polycyclic presentation:
2^2,3,2^6|a^2=b,f^2=h,g^2=i,c^a=c^2,d^a=de,f^a=fg,g^a=gi,h^a=hi,d^c=e,e^c=de,f^c=ghi,g^c=fgh,h^c=i,i^c=hi48 conjugacy classes — view the full character table.
| maximal subgroup of | #72210007, C2×#11714246, #72231237, … |
| maximal normal subgroup of | C2×#11714246, #72231237, #72238096, … |
| maximal quotient of | #72210007, #72231104, C2×#11714246, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1088627);