Finite Group DatabasebetaFinite Group Databasebeta| order | 32 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | Aut(G) = C23×D4 (order 64) · Inn(G) = D4 · Out(G) = C23 |
| holomorph | G ⋊ Aut(G) = #72219639 |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C22 | D4 |
| commutator | C4 | C2×C4 |
| Frattini, cosocle | C2×C4 | C22 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
21 conjugacy classes of subgroups.
| minimal degree | μ(G) = 12 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 2 generators on 12 points:
(1,2,3,4)(6,9)(7,11)(10,12);(5,6,8,10)(7,12,11,9)Polycyclic presentation:
2^5|a^2=d,b^2=e,c^2=e,b^a=bc,c^a=ce,c^b=ce14 conjugacy classes — view the full character table.
| Fitting subgroup of | #103×A5, #14514955, #14515039, … |
| maximal subgroup of | C2×#103, #416, #417, … |
| maximal normal subgroup of | C2×#103, #416, #417, … |
| maximal quotient of | Q8⋊C8, #328, #332, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32, 10);