Finite Group DatabasebetaFinite Group Databasebeta| order | 324 |
| isoclinism | isoclinic class of #4207 |
| automorphism | Aut(G) = C2×(#4207⋊C2)⋊C3 (order 1944) · Inn(G) = #4207 · Out(G) = C22×C3 |
| homology | Abelianization (H1) = C3×C4 · Schur multiplier (H2) = C3 |
No aliases for this group.
| derived | C3×C9 ⋊ C3×C4 |
| Sylow | #737 ⋊ C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | #4207 |
| commutator, hypocenter | C3×C9 | C3×C4 |
| Frattini, socle | C2×C3 | C3×C3⋊D3 |
| Fitting | C2×#737 | C2 |
| prime radical | C2×H(3,𝔽3) | D3 |
| cosocle | C2×C3×C9 | C2×C3 |
| LCS | G ▷ C3×C9 | |
| UCS | C1 ◁ C2 | |
63 conjugacy classes of subgroups.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 108 |
A minimal faithful permutation representation — 3 generators on 31 points:
(2,5)(3,19,21,9,10,20)(4,11)(6,8)(7,24,26,15,16,25)(12,27,14,17,22,23)(13,18)(28,29,30,31);(1,3,9)(2,7,17)(4,10,20)(5,12,15)(6,14,24)(8,16,23)(11,21,19)(13,22,25)(18,26,27);(1,18,5,11,8,6,4,2,13)(3,26,12,21,16,14,10,7,22)(9,27,15,19,23,24,20,17,25)Polycyclic presentation:
2,3,2,3^3|a^2=c,d^3=f^2,d^a=d^2f,e^a=e^2,f^a=f^2,e^b=ef42 conjugacy classes — view the full character table.
| maximal subgroup of | #10619905, #4207⋊C4, #10619908, … |
| maximal normal subgroup of | #10619905, #4207⋊C4, #10619908, … |
| maximal quotient of | #10619724, C2×#66409, #11757598, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(324, 101);