Finite Group DatabasebetaFinite Group Databasebeta| order | 72 |
| isoclinism | isoclinic class of Dic18 |
| automorphism | Aut(G) = C22×AGL(1,ℤ/9) (order 216) · Inn(G) = C2×D9 · Out(G) = C2×C3 |
| holomorph | G ⋊ Aut(G) = #72219992 |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Fitting | C22×C9 ⋊ C2 |
| Sylow | C9 ⋊ D4 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C2×D9 |
| commutator, cosocle | C2×C9 | C22 |
| Frattini, socle | C2×C3 | C2×D3 |
| Fitting | C22×C9 | C2 |
| hypercenter | C22 | D9 |
| hypocenter | C9 | D4 |
| LCS | G ▷ C2×C9 ▷ C9 | |
| UCS | C1 ◁ C2 ◁ C22 | |
24 conjugacy classes of subgroups.
| minimal degree | μ(G) = 13 |
| transitive degree | μt(G) = 36 |
A minimal faithful permutation representation — 2 generators on 13 points:
(1,6,8,7,2,4,3,9,5)(11,13);(2,4)(3,7)(5,6)(8,9)(10,11)(12,13)Polycyclic presentation:
2^3,3^2|d^3=e^2,b^a=bc,d^a=d^2e,e^a=e^221 conjugacy classes — view the full character table.
| maximal subgroup of | #3697, D4×D9, #3699, … |
| maximal normal subgroup of | #3697, D4×D9, #3699, … |
| maximal quotient of | Dic9⋊C4, (C2×D9)⋊C4, #3672, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(72, 8);