Finite Group DatabasebetaFinite Group Databasebeta| order | 324 |
| isoclinism | isoclinic class of #4184 |
| automorphism | Aut(G) = C2×(#4176⋊C2)⋊C3 (order 1944) · Inn(G) = #4184 · Out(G) = C22×C3 |
| homology | Abelianization (H1) = C4 · Schur multiplier (H2) = C1 |
No aliases for this group.
| derived, Sylow | #731 ⋊ C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | #4184 |
| commutator, hypocenter | #731 | C4 |
| Frattini | C2×C32 | C3⋊D3 |
| Fitting, cosocle | C2×#731 | C2 |
| socle | C2×C3 | #272 |
| prime radical | C2×H(3,𝔽3) | D3 |
| LCS | G ▷ #731 | |
| UCS | C1 ◁ C2 | |
42 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)(6,10)(7,13)(8,9)(11,21)(12,23)(14,26)(15,16)(17,25)(18,20)(19,29)(22,31)(24,27)(28,30);(5,17,13,9,16,15,8,7,25)(6,14,29,12,28,22,18,11,24)(10,27,21,20,31,30,23,19,26);(5,6,10)(7,14,27)(8,18,23)(9,12,20)(11,19,16)(13,24,26)(15,29,21)(17,28,31)(22,30,25)Polycyclic presentation:
2^2,3^4|a^2=b,d^3=f,c^a=c^2,d^a=d^2f^2,e^a=ef^2,f^a=f^2,d^c=de,e^c=ef26 conjugacy classes — view the full character table.
| maximal subgroup of | #4176⋊C4, #4177⋊C4, #10619557, … |
| maximal normal subgroup of | #4176⋊C4, #4177⋊C4, #10619557, … |
| maximal quotient of | #10619510, C2×#66331, #11757322, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(324, 23);