Finite Group DatabasebetaFinite Group Databasebeta| order | 672 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Fitting, Hall | C7×H(5,𝔽2) ⋊ C3 |
| Sylow | H(5,𝔽2) ⋊ ΓL(1,𝔽8) |
| Sylow | C7 ⋊ #994 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C22×C22⋊ΓL(1,𝔽8) |
| commutator, cosocle, hypocenter | C7×Q8 | C22×C3 |
| Fitting | C7×H(5,𝔽2) | C3 |
| socle | C2×C7 | C22×A4 |
| hypercenter | Q8 | C22⋊ΓL(1,𝔽8) |
| LCS | G ▷ C7×Q8 | |
| UCS | C1 ◁ C2 ◁ Q8 | |
92 conjugacy classes of subgroups.
| transitive degree | μt(G) = 168 |
Polycyclic presentation:
2^2,3,2^2,7,2|a^2=g,b^2=g,d^2=g,e^2=g,b^a=bg,d^c=e,e^c=deg,f^c=f^2,e^d=eg53 conjugacy classes — view the full character table.
| maximal subgroup of | #13057682, #13057683, #576⋊ΓL(1,𝔽8), … |
| maximal normal subgroup of | #13057682, #13057683, #576⋊ΓL(1,𝔽8), … |
| maximal quotient of | #520⋊ΓL(1,𝔽8), C4⋊(#41⋊ΓL(1,𝔽8)), Q8×C7⋊SL(2,𝔽3), … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(672, 1197);