Finite Group DatabasebetaFinite Group Databasebeta| order | 672 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Fitting | C7×#41 ⋊ C2×C3 |
| Hall | #6554 ⋊ C3 |
| Sylow | C7 ⋊ 2−1+4⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C2×D7⋊A4 |
| commutator, cosocle, hypocenter | C7×Q8 | C22×C3 |
| Fitting | C7×#41 | C2×C3 |
| socle | C2×C7 | C22×A4 |
| hypercenter | C4 | D7⋊A4 |
| LCS | G ▷ C7×Q8 | |
| UCS | C1 ◁ C2 ◁ C4 | |
92 conjugacy classes of subgroups.
| transitive degree | μt(G) = 112 |
Polycyclic presentation:
2^2,3,2^2,7,2|b^2=g,d^2=g,e^2=g,b^a=bg,f^a=f^6,d^c=e,e^c=deg,f^c=f^4,e^d=eg29 conjugacy classes — view the full character table.
| maximal subgroup of | #13057659, #13057660, #13057682, … |
| maximal normal subgroup of | #13057659, #13057660, #13057682, … |
| maximal quotient of | #13057639, (D7⋊SL(2,𝔽3))⋊C4, C4⋊(D7⋊SL(2,𝔽3)), … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(672, 1123);