Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| derived | #5465 ⋊ C4 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #5934 |
| commutator, hypocenter | #5465 | C4 |
| Frattini, hypercenter | C23 | #1020 |
| Fitting | C2×H(3,𝔽4) | D3 |
| prime radical, cosocle | C2×#5465 | C2 |
| LCS | G ▷ #5465 | |
| UCS | C1 ◁ C22 ◁ C23 | |
350 conjugacy classes of subgroups; only the 19 normal subgroups are shown.
This group has 350 conjugacy classes of subgroups — too many to draw. Only its 19 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 32 |
Polycyclic presentation:
2^2,3,2^6|a^2=b,c^a=c^2,d^a=de,f^a=fg,h^a=hi,d^c=e,e^c=de,f^c=g,g^c=fg,f^d=fi,g^d=ghi,f^e=fh,g^e=gi32 conjugacy classes — view the full character table.
| maximal subgroup of | #72227615, #72227700, C4×#85557, … |
| maximal normal subgroup of | #72227615, C4×#85557, #72261182 |
| maximal quotient of | #72227693, #72242927, #72258524 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1088653);