Finite Group DatabasebetaFinite Group Databasebeta| order | 72 |
| isoclinism | isoclinic class of #625 |
| automorphism | Aut(G) = C22×D32 (order 144) · Inn(G) = D32 · Out(G) = C22 |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Sylow | C32 ⋊ D4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | D32 |
| commutator, Fitting, socle, cosocle | C2×C32 | C22 |
| hypocenter | C32 | D4 |
| LCS | G ▷ C2×C32 ▷ C32 | |
| UCS | C1 ◁ C2 | |
37 conjugacy classes of subgroups.
| minimal degree | μ(G) = 10 |
| transitive degree | μt(G) = 12 |
A minimal faithful permutation representation — 2 generators on 10 points:
(1,2,3,4)(6,7)(8,9,10);(2,4)(5,6,7)(9,10)Polycyclic presentation:
2^3,3^2|b^2=c,b^a=bc,e^a=e^2,d^b=d^215 conjugacy classes — view the full character table.
| maximal subgroup of | #3796, #3798, #3799, … |
| maximal normal subgroup of | #3796, #3798, #3799, … |
| maximal quotient of | #3714, #3716, #3717, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(72, 23);