Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting | #3407 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×#5402 |
| commutator, cosocle | #131⋊C3 | C23 |
| Frattini | C8 | C22×S4 |
| Fitting | #3407 | D3 |
| hypercenter | M32 | S4 |
| hypocenter | SL(2,𝔽3) | C2×D8 |
| LCS | G ▷ #131⋊C3 ▷ #41⋊C3 ▷ SL(2,𝔽3) | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ M32 | |
359 conjugacy classes of subgroups; only the 45 normal subgroups are shown.
This group has 359 conjugacy classes of subgroups — too many to draw. Only its 45 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
Polycyclic presentation:
2^5,3,2^3|c^2=de,d^2=ei,e^2=i,g^2=i,h^2=i,c^a=cd,d^a=de,e^a=ei,f^a=f^2,g^a=gh,h^a=hi,c^b=ci,g^f=h,h^f=ghi,h^g=hi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1088579);