Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | #2202 ⋊ D3 |
| derived | #5465 ⋊ C22 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #73026 |
| commutator, cosocle, hypocenter | #5465 | C22 |
| Frattini | C2×Q8 | C2×S4 |
| Fitting | #2202 | D3 |
| hypercenter | C22 | #1020⋊C2 |
| LCS | G ▷ #5465 | |
| UCS | C1 ◁ C2 ◁ C22 | |
237 conjugacy classes of subgroups; only the 12 normal subgroups are shown.
This group has 237 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.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 16 |
Polycyclic presentation:
2^2,3,2^6|f^2=i,g^2=i,c^a=c^2,d^a=de,f^a=fg,g^a=gi,h^a=hi,d^b=dfhi,e^b=eghi,h^b=hi,d^c=e,e^c=de,f^c=g,g^c=fgi,f^d=fi,g^d=ghi,f^e=fh,g^e=gi,g^f=gi23 conjugacy classes — view the full character table.
| maximal subgroup of | #72209911, #72241408 |
| maximal normal subgroup of | #72241408 |
| maximal quotient of | #72227859, #72228025, #72228075, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1086052);