Finite Group DatabasebetaFinite Group Databasebeta| order | 72 |
| isoclinism | isoclinic class of #157 |
| automorphism | Aut(G) = C2×AΓL(1,𝔽9) (order 288) · Inn(G) = #157 · Out(G) = C23 |
| holomorph | G ⋊ Aut(G) = #72220311 |
| homology | Abelianization (H1) = C8 · Schur multiplier (H2) = C3 |
No aliases for this group.
| derived, Sylow | C32 ⋊ C8 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | #157 |
| commutator, hypocenter | C32 | C8 |
| Fitting, socle, prime radical | C2×C32 | C4 |
| cosocle | C3⋊Dic3 | C2 |
| LCS | G ▷ C32 | |
| UCS | C1 ◁ C2 | |
14 conjugacy classes of subgroups.
| minimal degree | μ(G) = 14 |
| transitive degree | μt(G) = 24 |
A minimal faithful permutation representation — 2 generators on 14 points:
(1,2)(3,5,6,4)(7,8,9,11,10,12,13,14);(1,3,6)(2,5,4)Polycyclic presentation:
2^3,3^2|a^2=b,b^2=c,d^a=de^2,e^a=d^2e^2,d^b=d^2,e^b=e^2| C1 | C2 | C3 | C4 | C5 | C6 | C7 | C8 | C9 | C10 | C11 | C12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| size | 1 | 1 | 4 | 4 | 4 | 4 | 9 | 9 | 9 | 9 | 9 | 9 |
| order | 1 | 2 | 3 | 3 | 6 | 6 | 4 | 4 | 8 | 8 | 8 | 8 |
| χ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| χ2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 |
| χ3 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | ||||
| χ4 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | ||||
| χ5 | 1 | -1 | 1 | 1 | -1 | -1 | ζ | −ζ | ζ3 | −ζ3 | ||
| χ6 | 1 | -1 | 1 | 1 | -1 | -1 | −ζ | ζ | −ζ3 | ζ3 | ||
| χ7 | 1 | -1 | 1 | 1 | -1 | -1 | ζ3 | −ζ3 | ζ | −ζ | ||
| χ8 | 1 | -1 | 1 | 1 | -1 | -1 | −ζ3 | ζ3 | −ζ | ζ | ||
| χ9 | 4 | 4 | 1 | -2 | 1 | -2 | 0 | 0 | 0 | 0 | 0 | 0 |
| χ10 | 4 | 4 | -2 | 1 | -2 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| χ11 | 4 | -4 | 1 | -2 | -1 | 2 | 0 | 0 | 0 | 0 | 0 | 0 |
| χ12 | 4 | -4 | -2 | 1 | 2 | -1 | 0 | 0 | 0 | 0 | 0 | 0 |
Here ζ is a primitive root of unity of order 8 (ζ8 = 1); i = ζ2.
| derived subgroup of | #10593160, #10593161, #10593162, … |
| maximal subgroup of | #3771, #3774, #3775, … |
| maximal normal subgroup of | #3771, #3774, #3775, … |
| maximal quotient of | #3708, C2×#622, #6188, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(72, 19);