Finite Group DatabasebetaFinite Group Databasebeta| order | 324 |
| isoclinism | isoclinic class of #4185 |
| automorphism | ? |
| homology | Abelianization (H1) = C4 · Schur multiplier (H2) = C3 |
No aliases for this group.
| derived, Sylow | #732 ⋊ C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | #4185 |
| commutator, hypocenter | #732 | C4 |
| Frattini | C2×C32 | C3⋊D3 |
| Fitting, prime radical, cosocle | C2×#732 | C2 |
| socle | C2×C3 | #272 |
| LCS | G ▷ #732 | |
| UCS | C1 ◁ C2 | |
48 conjugacy classes of subgroups.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 108 |
A minimal faithful permutation representation — 3 generators on 31 points:
(1,2,3,4)(6,9)(7,13)(8,15)(10,18)(11,17)(12,22)(14,25)(16,20)(19,24)(21,30)(23,29)(26,27)(28,31);(5,7,13)(6,10,20)(8,14,26)(9,16,18)(11,23,31)(12,21,24)(15,27,25)(17,28,29)(19,30,22);(5,6,9)(7,10,19)(8,12,17)(11,22,15)(13,24,18)(14,21,16)(20,30,25)(23,28,27)(26,31,29)Polycyclic presentation:
2^2,3^4|a^2=b,c^a=c^2,d^a=d^2,f^a=f^2,d^c=de,e^c=ef,e^d=ef^226 conjugacy classes — view the full character table.
| maximal subgroup of | #4178⋊C4, #4179⋊C4, #10619564, … |
| maximal normal subgroup of | #4178⋊C4, #4179⋊C4, #10619564, … |
| maximal quotient of | #10619511, C2×#66332, #11757321, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(324, 24);