Finite Group DatabasebetaFinite Group Databasebeta| order | 32 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | Aut(G) = C22×S4 (order 96) · Inn(G) = C22 · Out(G) = C22×D3 |
| holomorph | G ⋊ Aut(G) = #72215818 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C8 | C22 |
| commutator, socle | C2 | C22×C4 |
| Frattini, cosocle | C4 | C23 |
| prime radical | #41 | C2 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ G | |
| LECS | G ▷ C4 ▷ C2 ▷ C1 | |
31 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 16 |
A minimal faithful permutation representation — 3 generators on 16 points:
(1,2,4,7,5,8,11,14)(3,6,10,13,9,12,15,16);(1,3)(2,6)(4,10)(5,9)(7,13)(8,12)(11,15)(14,16);(3,9)(6,12)(10,15)(13,16)Polycyclic presentation:
2^5|a^2=d,d^2=e,c^b=ce20 conjugacy classes — view the full character table.
| derived subgroup of | #3415⋊C3, #3416⋊C3, #3417⋊C3, … |
| Fitting subgroup of | #131⋊C3, #4628, #5403, … |
| maximal subgroup of | #350, #443, #444, … |
| maximal normal subgroup of | #350, #443, #444, … |
| maximal quotient of | #405, #408, #424, … |
| outer automorphism group of | #13648, D5⋊C32, C2×#3928 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32, 38);