Finite Group DatabasebetaFinite Group Databasebeta| order | 72 |
| isoclinism | isoclinic class of #625 |
| automorphism | Aut(G) = #64241 (order 288) · Inn(G) = D32 · Out(G) = D4 |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Sylow | C32 ⋊ Q8 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | D32 |
| commutator, Fitting, socle, prime radical, cosocle | C2×C32 | C22 |
| hypocenter | C32 | Q8 |
| LCS | G ▷ C2×C32 ▷ C32 | |
| UCS | C1 ◁ C2 | |
27 conjugacy classes of subgroups.
| minimal degree | μ(G) = 14 |
| transitive degree | μt(G) = 24 |
A minimal faithful permutation representation — 2 generators on 14 points:
(1,3,4,8)(2,5,6,7)(10,11)(12,13,14);(1,2,4,6)(3,7,8,5)(9,10,11)(13,14)Polycyclic presentation:
2^3,3^2|a^2=c,b^2=c,b^a=bc,e^a=e^2,d^b=d^215 conjugacy classes — view the full character table.
| maximal subgroup of | #3775, #3776, D3×Dic6, … |
| maximal normal subgroup of | #3775, #3776, D3×Dic6, … |
| maximal quotient of | Dic3⋊Dic3, #3724, C2×#627, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(72, 24);