Finite Group DatabasebetaFinite Group Databasebeta| order | 32 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | Aut(G) = #87524 (order 384) · Inn(G) = C22 · Out(G) = GL(2,ℤ/4) |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, prime radical, cosocle | C23 | C22 |
| commutator | C2 | C42 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
38 conjugacy classes of subgroups.
| minimal degree | μ(G) = 12 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 2 generators on 12 points:
(6,8)(9,10,12,11);(1,2,3,4)(5,6,7,8)(9,11,12,10)Polycyclic presentation:
2^5|a^2=d,b^2=e,b^a=bc20 conjugacy classes — view the full character table.
| derived subgroup of | #95⋊C3, C2×#95⋊C3, #376⋊C3, … |
| Fitting subgroup of | #95⋊C3, #4621, #4622, … |
| maximal subgroup of | #327, #328, C2×#95, … |
| maximal normal subgroup of | #327, #328, C2×#95, … |
| maximal quotient of | #336, H(3,ℤ/4), #338, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32, 2);