Finite Group DatabasebetaFinite Group Databasebeta| order | 324 |
| automorphism | Aut(G) = C3×A4⋊#272 (order 1944) · Inn(G) = C32×A4 · Out(G) = C3×D3 |
| homology | Abelianization (H1) = C33 · Schur multiplier (H2) = C2×C32 |
No aliases for this group.
| Fitting | C22×3−1+2 ⋊ C3 |
| Sylow | C22 ⋊ #737 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C3 | C32×A4 |
| commutator, socle, cosocle | C22×C3 | C33 |
| Fitting | C22×3−1+2 | C3 |
| prime radical | C22⋊H(3,𝔽3) | C3 |
| hypercenter | 3−1+2 | A4 |
| hypocenter | C22 | #737 |
| LCS | G ▷ C22×C3 ▷ C22 | |
| UCS | C1 ◁ C3 ◁ 3−1+2 | |
74 conjugacy classes of subgroups.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 54 |
A minimal faithful permutation representation — 3 generators on 31 points:
(1,4)(2,3)(6,11,12)(7,14,15)(9,18,21)(10,20,22)(13,23,24)(25,30,31);(5,7,13,8,15,23,16,14,24)(6,10,19,12,20,27,11,22,29)(9,17,25,18,26,31,21,28,30);(2,3,4)(5,6,9)(7,10,17)(8,12,18)(11,21,16)(13,19,25)(14,22,28)(15,20,26)(23,27,31)(24,29,30)Polycyclic presentation:
3^4,2^2|b^3=d,c^a=cd,e^a=f,f^a=ef,c^b=cd44 conjugacy classes — view the full character table.
| maximal subgroup of | #66440⋊C2, C2×#66440, #11757798, … |
| maximal normal subgroup of | #66440⋊C2, C2×#66440, #11757798, … |
| maximal quotient of | #10619916, C2×#66440, #11757766, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(324, 132);