Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | #2897 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | D4×S4 |
| commutator, cosocle | C2×#41⋊C3 | C23 |
| Frattini | C2×C4 | C22×S4 |
| Fitting | #2897 | D3 |
| hypercenter | #103 | S4 |
| hypocenter | SL(2,𝔽3) | #120 |
| LCS | G ▷ C2×#41⋊C3 ▷ SL(2,𝔽3) | |
| UCS | C1 ◁ C22 ◁ C2×C4 ◁ #103 | |
715 conjugacy classes of subgroups; only the 51 normal subgroups are shown.
This group has 715 conjugacy classes of subgroups — too many to draw. Only its 51 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:
2^5,3,2^3|b^2=i,c^2=e,d^2=i,g^2=i,h^2=i,c^a=ce,f^a=f^2,g^a=gh,h^a=hi,c^b=cd,d^b=di,d^c=di,g^f=h,h^f=ghi,h^g=hi47 conjugacy classes — view the full character table.
| maximal subgroup of | #72225981, #72226217, #72226229, … |
| maximal normal subgroup of | #72226217, #72226229, #72226271 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1087781);