Finite Group DatabasebetaFinite Group Databasebeta| order | 324 |
| isoclinism | isoclinic class of #66343 |
| automorphism | ? |
| homology | Abelianization (H1) = C4 · Schur multiplier (H2) = C9 |
No aliases for this group.
| Fitting, derived, Sylow | C92 ⋊ C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, Fitting, hypocenter | C92 | C4 |
| Frattini, socle | C32 | #157 |
| prime radical, cosocle | C9⋊D9 | C2 |
| LCS | G ▷ C92 | |
| UCS | C1 | |
29 conjugacy classes of subgroups.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 18 |
A minimal faithful permutation representation — 2 generators on 18 points:
(1,2)(3,6,8,5)(4,11,12,7)(9,14,10,13)(15,16,18,17);(1,3,9,12,17,16,4,10,8)(2,6,14,7,15,18,11,13,5)Polycyclic presentation:
2^2,3^4|a^2=b,c^3=e^2,d^3=f,c^a=cd^2,d^a=c^2d^2ef^2,e^a=ef,f^a=ef^2,c^b=c^2e,d^b=d^2f^2,e^b=e^2,f^b=f^224 conjugacy classes — view the full character table.
| derived subgroup of | #10619732⋊C2 |
| maximal subgroup of | #10619732, #66343⋊C2, #10619734, … |
| maximal normal subgroup of | #10619732, #66343⋊C2, #10619734, … |
| maximal quotient of | #10619526, C2×#66343, #11757369, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(324, 35);