Finite Group DatabasebetaFinite Group Databasebeta| order | 32 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | Aut(G) = C22×#120 (order 128) · Inn(G) = C23 · Out(G) = C24 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
| unambiguous semidirect product | C22⋊Q8 |
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C22 | C23 |
| prime radical | C23 | C22 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ G | |
| LECS | G ▷ C22 ▷ C1 | |
37 conjugacy classes of subgroups.
| minimal degree | μ(G) = 12 |
| transitive degree | μt(G) = 16 |
A minimal faithful permutation representation — 3 generators on 12 points:
(2,4)(5,6,8,10)(7,11,12,9);(5,7,8,12)(6,9,10,11);(1,2)(3,4)(5,6)(7,9)(8,10)(11,12)Polycyclic presentation:
2^5|a^2=d,b^2=d,b^a=bd,c^a=ce14 conjugacy classes — view the full character table.
| Fitting subgroup of | C22⋊Q8×A5, C22⋊#91883, #14515209, … |
| maximal subgroup of | #328, #450, #451, … |
| maximal normal subgroup of | #328, #450, #451, … |
| maximal quotient of | #380, #385, #387, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32, 29);