Finite Group DatabasebetaFinite Group Databasebeta| order | 672 |
| isoclinism | isoclinic class of D3×D7 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C8 · Schur multiplier (H2) = C4 |
No aliases for this group.
| derived, Hall | C3×C7 ⋊ C4×C8 |
| Sylow | C7 ⋊ C8×Dic3 |
| Sylow | C3 ⋊ C8×Dic7 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2×C4 | D3×D7 |
| commutator, hypocenter | C3×C7 | C4×C8 |
| Fitting, cosocle | C2×C3×C4×C7 | C22 |
| socle, prime radical | C22×C3×C7 | C2×C4 |
| LCS | G ▷ C3×C7 | |
| UCS | C1 ◁ C2×C4 | |
88 conjugacy classes of subgroups.
| transitive degree | μt(G) = 672 |
Polycyclic presentation:
2^5,3,7|a^2=c,b^2=cd,c^2=e,g^a=g^6,f^b=f^2120 conjugacy classes — view the full character table.
| maximal subgroup of | #13052229, #13052230, #13052233, … |
| maximal normal subgroup of | #13052229, #13052230, #13052233, … |
| maximal quotient of | C3⋊C8×C7⋊C8, #13050861, #13050900, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(672, 68);