Finite Group DatabasebetaFinite Group Databasebeta| order | 504 |
| isoclinism | isoclinic class of #92722 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C3 |
No aliases for this group.
| Hall | C32×C7 ⋊ Q8 |
| Sylow | C32 ⋊ Dic14 |
| Sylow | C7 ⋊ #634 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C2×C3⋊D21 |
| commutator, socle, prime radical, cosocle | C2×C32×C7 | C22 |
| Fitting | C32×C4×C7 | C2 |
| hypercenter | C4 | C3⋊D21 |
| hypocenter | C32×C7 | Q8 |
| LCS | G ▷ C2×C32×C7 ▷ C32×C7 | |
| UCS | C1 ◁ C2 ◁ C4 | |
72 conjugacy classes of subgroups.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 504 |
A minimal faithful permutation representation — 3 generators on 21 points:
(1,2,4,6,7,5,3)(8,10,11,15)(9,12,13,14)(19,20,21);(2,3)(4,5)(6,7)(8,9,11,13)(10,14,15,12)(17,18)(20,21);(16,17,18)Polycyclic presentation:
2^3,3^2,7|a^2=c,b^2=c,b^a=bc,d^a=d^2,e^a=e^2,f^a=f^6129 conjugacy classes — view the full character table.
| maximal subgroup of | #11759158, #11759160, #11759163, … |
| maximal normal subgroup of | #11759158, #11759160, #11759163, … |
| maximal quotient of | #11759311, C4⋊(C3⋊Dic21), C2×#92722, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(504, 148);