Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting | C22×2−1+4 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #87524 |
| commutator, cosocle | #5948 | C22 |
| Frattini | C4 | C2×#1020 |
| Fitting | C22×2−1+4 | D3 |
| socle | C23 | GL(2,ℤ/4) |
| hypercenter | D4 | #1020 |
| hypocenter | C22⋊SL(2,𝔽3) | D4 |
| LCS | G ▷ #5948 ▷ C22⋊SL(2,𝔽3) | |
| UCS | C1 ◁ C2 ◁ C4 ◁ D4 | |
889 conjugacy classes of subgroups; only the 23 normal subgroups are shown.
This group has 889 conjugacy classes of subgroups — too many to draw. Only its 23 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:
2^3,3,2^5|c^2=i,g^2=i,h^2=i,b^a=bc,c^a=ci,d^a=d^2,e^a=ef,g^a=gh,h^a=hi,c^b=ci,e^d=f,f^d=ef,g^d=h,h^d=ghi,h^g=hi33 conjugacy classes — view the full character table.
| maximal subgroup of | #72211581, #72227573, C2×#11715702, … |
| maximal normal subgroup of | #72227573, C2×#11715702, #72268583 |
| maximal quotient of | #72211581, #72227571, C2×#11715702, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1090083);