Finite Group DatabasebetaFinite Group Databasebeta| order | 672 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Fitting | C7×#41 ⋊ D3 |
| Sylow | C7 ⋊ #985 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C4 | C7⋊S4 |
| commutator, cosocle, hypocenter | C7×SL(2,𝔽3) | C22 |
| Frattini | C2 | C2×C7⋊S4 |
| Fitting | C7×#41 | D3 |
| socle | C2×C7 | C2×S4 |
| LCS | G ▷ C7×SL(2,𝔽3) | |
| UCS | C1 ◁ C4 | |
78 conjugacy classes of subgroups.
| transitive degree | μt(G) = 112 |
Polycyclic presentation:
2^2,3,7,2^3|b^2=g,e^2=g,f^2=g,c^a=c^2,d^a=d^6,e^a=fg,f^a=eg,e^c=f,f^c=efg,f^e=fg58 conjugacy classes — view the full character table.
| maximal subgroup of | #13057079, #13057081, #13057172, … |
| maximal normal subgroup of | #13057079, #13057081, #13057172, … |
| maximal quotient of | C4×C7⋊#226, #13057158, C4×C7⋊GL(2,𝔽3), … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(672, 1085);