Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C2 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting | C22×#95 ⋊ D3 |
| derived | C22⋊(#95⋊C3) ⋊ C2 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | #85547 |
| commutator, cosocle, hypocenter | C22⋊(#95⋊C3) | C2 |
| Frattini | C23 | #1020 |
| Fitting | C22×#95 | D3 |
| socle | C25 | S4 |
| LCS | G ▷ C22⋊(#95⋊C3) | |
| UCS | C1 ◁ C2 | |
401 conjugacy classes of subgroups; only the 15 normal subgroups are shown.
This group has 401 conjugacy classes of subgroups — too many to draw. Only its 15 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:
2,3,2^7|e^2=gi,f^2=ghi,b^a=b^2,c^a=cd,e^a=ef,f^a=fghi,g^a=gi,h^a=hi,c^b=cd,d^b=c,e^b=efg,f^b=e,g^b=ghi,h^b=i,i^b=hi,f^e=fgh34 conjugacy classes — view the full character table.
| maximal subgroup of | #72226656, #72228154, #72228155, … |
| maximal normal subgroup of | #72226656, C2×#11714237, #72231199, … |
| maximal quotient of | #72231104, C2×#11714237, #72258356, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1088618);