Finite Group DatabasebetaFinite Group Databasebeta| order | 324 |
| isoclinism | isoclinic class of #4174 |
| automorphism | Aut(G) = C2×C3×((C3⋊D3)⋊C3)⋊C2 (order 648) · Inn(G) = (C3⋊D3)⋊C3 · Out(G) = C22×C3 |
| homology | Abelianization (H1) = C3×C4 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Sylow | #731 ⋊ C4 |
| subgroup | G/H | |
|---|---|---|
| center, socle, hypercenter | C2×C3 | (C3⋊D3)⋊C3 |
| commutator, hypocenter | H(3,𝔽3) | C3×C4 |
| Frattini | C2×C32 | C3×D3 |
| Fitting | C2×#731 | C2 |
| prime radical, cosocle | C2×H(3,𝔽3) | C2×C3 |
| LCS | G ▷ H(3,𝔽3) | |
| UCS | C1 ◁ C2×C3 | |
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:
(1,18,14,4,2,24,12,7,5)(3,26,16,20,9,23,10,15,21,8,17,25,11,19,13,22,6,27)(28,29,30,31);(1,3,8)(2,6,15)(4,11,20)(5,13,23)(7,17,26)(9,19,18)(10,22,12)(14,21,27)(16,25,24)Polycyclic presentation:
2,3,2,3^3|a^2=c,b^3=f,d^a=d^2,e^a=e^2f^2,d^b=de,e^d=ef^244 conjugacy classes — view the full character table.
| maximal subgroup of | #4184⋊C4, #4177⋊C4, #10619558, … |
| maximal normal subgroup of | #4184⋊C4, #4177⋊C4, #10619558, … |
| maximal quotient of | #10619502, C2×#66323, #11757285, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(324, 15);