Finite Group DatabasebetaFinite Group Databasebeta| order | 72 |
| isoclinism | isoclinic class of #634 |
| automorphism | Aut(G) = D4×AGL(2,𝔽3) (order 3456) · Inn(G) = C2×C3⋊D3 · Out(G) = C2×GL(2,𝔽3) |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C3 |
No aliases for this group.
| Sylow | C32 ⋊ Q8 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C2×C3⋊D3 |
| commutator, socle, prime radical, cosocle | C2×C32 | C22 |
| Fitting | C32×C4 | C2 |
| hypercenter | C4 | C3⋊D3 |
| hypocenter | C32 | Q8 |
| LCS | G ▷ C2×C32 ▷ C32 | |
| UCS | C1 ◁ C2 ◁ C4 | |
36 conjugacy classes of subgroups.
| minimal degree | μ(G) = 14 |
| transitive degree | μt(G) = 72 |
A minimal faithful permutation representation — 3 generators on 14 points:
(1,3,4,8)(2,5,6,7)(12,13,14);(1,2,4,6)(3,7,8,5)(10,11)(13,14);(9,10,11)Polycyclic presentation:
2^3,3^2|a^2=c,b^2=c,b^a=bc,d^a=d^2,e^a=e^221 conjugacy classes — view the full character table.
| maximal subgroup of | #3716, #3719, #3744, … |
| maximal normal subgroup of | #3716, #3719, #3744, … |
| maximal quotient of | #3750, C4⋊(C3⋊Dic3), C2×#634, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(72, 31);