Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting | #2029 ⋊ D3 |
| derived | #543⋊C3 ⋊ C22 |
| subgroup | G/H | |
|---|---|---|
| center, socle, hypercenter | C22 | #1020⋊C2 |
| commutator, cosocle, hypocenter | #543⋊C3 | C22 |
| Frattini | C24 | C2×S4 |
| Fitting | #2029 | D3 |
| LCS | G ▷ #543⋊C3 | |
| UCS | C1 ◁ C22 | |
572 conjugacy classes of subgroups; only the 17 normal subgroups are shown.
This group has 572 conjugacy classes of subgroups — too many to draw. Only its 17 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 16 |
Polycyclic presentation:
2^2,3,2^6|d^2=h,e^2=h,c^a=c^2,d^a=de,e^a=eh,f^a=fg,d^b=dfi,e^b=egi,d^c=e,e^c=deh,f^c=g,g^c=fg,e^d=eh,g^d=gi,f^e=fi33 conjugacy classes — view the full character table.
| maximal subgroup of | #72216969, #72218463, #72227813, … |
| maximal normal subgroup of | #72218463, #72227813, #72228090, … |
| maximal quotient of | #72241378, #72241402, C2×#11711401 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1085782);