Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C2 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | #85547 |
| commutator, prime radical, cosocle, hypocenter | #73286 | C2 |
| Frattini, socle | C23 | #1020 |
| Fitting | #2276 | D3 |
| LCS | G ▷ #73286 | |
| UCS | C1 ◁ C2 | |
191 conjugacy classes of subgroups; only the 12 normal subgroups are shown.
This group has 191 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) = 48 |
Polycyclic presentation:
2,3,2^7|a^2=g,c^2=hi,d^2=i,e^2=g,f^2=g,b^a=b^2,c^a=cd,d^a=di,e^a=ef,f^a=fg,h^a=hi,c^b=cd,d^b=ci,e^b=efg,f^b=e,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 | C2×#11714240, #72239183, #72239368, … |
| maximal normal subgroup of | C2×#11714240, #72239183, #72239368, … |
| maximal quotient of | #72238916, C2×#11714240 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1088621);