Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C2 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| Fitting | #2273 ⋊ D3 |
| derived | #2273⋊C3 ⋊ C2 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | #85547 |
| commutator, cosocle, hypocenter | #2273⋊C3 | C2 |
| Frattini, socle | C23 | #1020 |
| Fitting | #2273 | D3 |
| LCS | G ▷ #2273⋊C3 | |
| UCS | C1 ◁ C2 | |
310 conjugacy classes of subgroups; only the 12 normal subgroups are shown.
This group has 310 conjugacy classes of subgroups — too many to draw. Only its 12 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=ghi,f^2=gi,b^a=b^2,c^a=cd,e^a=ef,f^a=fgi,h^a=hi,c^b=cd,d^b=c,e^b=efg,f^b=ei,h^b=i,i^b=hi,f^c=fg,e^d=eg,f^e=fg29 conjugacy classes — view the full character table.
| maximal subgroup of | #72209658, #72209915, C2×#11714243, … |
| maximal normal subgroup of | #72209658, #72209915, C2×#11714243, … |
| maximal quotient of | #72209868, #72228154, C2×#11714243, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1088624);