Finite Group DatabasebetaFinite Group Databasebeta| order | 672 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Sylow | C7 ⋊ #984 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C2×C7⋊S4 |
| commutator, cosocle, hypocenter | C7×SL(2,𝔽3) | C22 |
| Fitting | C7×#41 | D3 |
| socle | C2×C7 | C2×S4 |
| prime radical | C7×#41⋊C3 | C2 |
| hypercenter | C4 | C7⋊S4 |
| LCS | G ▷ C7×SL(2,𝔽3) | |
| UCS | C1 ◁ C2 ◁ C4 | |
72 conjugacy classes of subgroups.
| transitive degree | μt(G) = 224 |
Polycyclic presentation:
2^2,3,7,2^3|a^2=g,b^2=g,e^2=g,f^2=g,b^a=bg,c^a=c^2,d^a=d^6,e^a=fg,f^a=eg,e^c=f,f^c=efg,f^e=fg55 conjugacy classes — view the full character table.
| maximal subgroup of | #13057082, #13057085, #13057091, … |
| maximal normal subgroup of | #13057082, #13057085, #13057091, … |
| maximal quotient of | #13057155, C4⋊(C7⋊#226), #13057157, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(672, 1084);