Finite Group DatabasebetaFinite Group Databasebeta| order | 324 |
| isoclinism | isoclinic class of #4177 |
| automorphism | Aut(G) = C2×(#4176⋊C2)⋊C3 (order 1944) · Inn(G) = #4177 · Out(G) = C22×C3 |
| homology | Abelianization (H1) = C3×C4 · Schur multiplier (H2) = C3 |
No aliases for this group.
| derived | C3×C9 ⋊ C3×C4 |
| Sylow | #731 ⋊ C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | #4177 |
| commutator, hypocenter | C3×C9 | C3×C4 |
| Frattini | C2×C32 | C3×D3 |
| Fitting | C2×#731 | C2 |
| socle | C2×C3 | (C3⋊D3)⋊C3 |
| prime radical | C2×H(3,𝔽3) | D3 |
| cosocle | C2×C3×C9 | C2×C3 |
| LCS | G ▷ C3×C9 | |
| UCS | C1 ◁ C2 | |
34 conjugacy classes of subgroups.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 108 |
A minimal faithful permutation representation — 2 generators on 31 points:
(2,12,14,5,6,24)(3,21,23,9,10,22)(4,11)(7,27,13,18,19,16)(8,25,26,20,17,15)(28,29,30,31);(1,8,5,4,18,7,11,2,20)(3,17,13,10,27,15,23,6,24)(9,14,12,22,26,19,21,16,25)Polycyclic presentation:
2,3,2,3^3|a^2=c,d^3=f,d^a=d^2f^2,e^a=e^2,f^a=f^2,d^b=de,e^b=ef26 conjugacy classes — view the full character table.
| maximal subgroup of | #4176⋊C4, #4184⋊C4, #10619556, … |
| maximal normal subgroup of | #4176⋊C4, #4184⋊C4, #10619556, … |
| maximal quotient of | #10619503, C2×#66324, #11757284, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(324, 16);