Finite Group DatabasebetaFinite Group Databasebeta| order | 32 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | Aut(G) = C22×C22⋊A4 (order 192) · Inn(G) = C23 · Out(G) = C2×A4 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C2 |
No aliases for this group.
| 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 | |
30 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 16 |
A minimal faithful permutation representation — 3 generators on 16 points:
(2,6)(3,10)(8,15)(9,13)(11,14)(12,16);(1,2,7,13)(3,8,14,15)(4,6,5,9)(10,16,11,12);(1,3,4,11)(2,8,6,12)(5,10,7,14)(9,16,13,15)Polycyclic presentation:
2^5|b^2=de,c^2=d,b^a=bd,c^a=ce14 conjugacy classes — view the full character table.
| Fitting subgroup of | #126⋊C3, #126×A5, #14515271 |
| maximal subgroup of | C2×#126, #529, #538, … |
| maximal normal subgroup of | C2×#126, #529, #538, … |
| maximal quotient of | #383, #387, #388, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32, 33);