Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting | #1779 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #72991 |
| commutator, cosocle | C4×C22⋊A4 | C22 |
| Frattini | C22×C4 | C2×S4 |
| Fitting | #1779 | D3 |
| socle | C23 | C4⋊S4 |
| hypercenter | C4 | #1020⋊C2 |
| hypocenter | C22⋊A4 | D8 |
| LCS | G ▷ C4×C22⋊A4 ▷ C2×C22⋊A4 ▷ C22⋊A4 | |
| UCS | C1 ◁ C2 ◁ C4 | |
446 conjugacy classes of subgroups; only the 18 normal subgroups are shown.
This group has 446 conjugacy classes of subgroups — too many to draw. Only its 18 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:
2^4,3,2^4|b^2=cd,c^2=d,b^a=bc,c^a=cd,e^a=e^2,f^a=fg,h^a=hi,f^b=fh,g^b=gi,f^e=g,g^e=fg,h^e=i,i^e=hi32 conjugacy classes — view the full character table.
| maximal subgroup of | #72252247, #72252248, #72252250, … |
| maximal normal subgroup of | #72252247, #72252248, #72252250, … |
| maximal quotient of | C2×#11711263, #72341962, #72342028 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1085644);