Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | C22×H(5,𝔽2) ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #85546 |
| commutator, cosocle | C23×A4 | C23 |
| Frattini | C23 | C22×S4 |
| Fitting | C22×H(5,𝔽2) | D3 |
| socle | C23 | #953 |
| hypercenter | H(5,𝔽2) | S4 |
| hypocenter | A4 | B(4,𝔽2) |
| LCS | G ▷ C23×A4 ▷ C2×A4 ▷ A4 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ H(5,𝔽2) | |
1340 conjugacy classes of subgroups; only the 59 normal subgroups are shown.
This group has 1340 conjugacy classes of subgroups — too many to draw. Only its 59 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 24 |
Polycyclic presentation:
2^6,3,2^2|b^a=be,c^a=cd,g^a=g^2,h^a=hi,d^b=df,e^c=ef,h^g=i,i^g=hi49 conjugacy classes — view the full character table.
| maximal subgroup of | #72219756, #72220596, S4×B(4,𝔽2), … |
| maximal normal subgroup of | S4×B(4,𝔽2), C2×#11713964, #72267053, … |
| maximal quotient of | #72219755, #72219756, #72220596, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1088345);