Finite Group DatabasebetaFinite Group Databasebeta| order | 672 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C4 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Hall | #6511 ⋊ C3 |
| Sylow | C7 ⋊ #131⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C4 | D7⋊A4 |
| commutator, hypocenter | C7×Q8 | C3×C4 |
| Fitting, cosocle | C7×#41 | C2×C3 |
| socle | C2×C7 | C4×A4 |
| prime radical | #41⋊ΓL(1,𝔽8) | C2 |
| LCS | G ▷ C7×Q8 | |
| UCS | C1 ◁ C4 | |
44 conjugacy classes of subgroups.
| transitive degree | μt(G) = 224 |
Polycyclic presentation:
2,3,2^3,7,2|a^2=c,c^2=g,d^2=g,e^2=g,f^a=f^6,d^b=e,e^b=deg,f^b=f^4,e^d=eg38 conjugacy classes — view the full character table.
| maximal subgroup of | D7⋊(#131⋊C3), #13057658, C2×#10620803, … |
| maximal normal subgroup of | D7⋊(#131⋊C3), #13057658, C2×#10620803, … |
| maximal quotient of | (C7⋊C8)⋊SL(2,𝔽3), C2×#10620803 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(672, 385);