Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | 2−1+6 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #85546 |
| commutator, cosocle | C2×#41⋊C3 | C23 |
| Frattini | C2×C4 | C22×S4 |
| Fitting | 2−1+6 | D3 |
| hypercenter | H(5,𝔽2) | S4 |
| hypocenter | SL(2,𝔽3) | #120 |
| LCS | G ▷ C2×#41⋊C3 ▷ SL(2,𝔽3) | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ H(5,𝔽2) | |
1003 conjugacy classes of subgroups; only the 57 normal subgroups are shown.
This group has 1003 conjugacy classes of subgroups — too many to draw. Only its 57 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
Polycyclic presentation:
2^5,3,2^3|e^2=i,g^2=i,h^2=i,b^a=be,c^a=cd,e^a=ei,f^a=f^2,g^a=gh,h^a=hi,d^b=di,e^b=ei,e^c=ei,g^f=h,h^f=ghi,h^g=hi47 conjugacy classes — view the full character table.
| maximal subgroup of | #72227648, #72228294, #72252447, … |
| maximal normal subgroup of | #72252447, #72258931, #72258947 |
| maximal quotient of | #72268142, #72268257, #72268987, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1088341);