Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2 | C4⋊#857 |
| commutator, cosocle | #376⋊C3 | C22 |
| Frattini | C22×C4 | C2×S4 |
| Fitting | #1726 | D3 |
| socle | C23 | C4⋊S4 |
| prime radical, hypocenter | #95⋊C3 | D4 |
| hypercenter | C8 | #857 |
| LCS | G ▷ #376⋊C3 ▷ #95⋊C3 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 | |
184 conjugacy classes of subgroups; only the 17 normal subgroups are shown.
This group has 184 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) = 96 |
Polycyclic presentation:
2^3,3,2^5|a^2=gh,b^2=cgh,c^2=gh,e^2=g,f^2=ghi,b^a=bc,c^a=cgh,d^a=d^2,e^a=efi,f^a=fghi,g^a=gi,h^a=hi,e^d=fh,f^d=efgh,g^d=ghi,h^d=i,i^d=hi,f^e=fgh36 conjugacy classes — view the full character table.
| maximal subgroup of | #72210808 |
| maximal quotient of | #72210769, #72210779 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1085610);