Finite Group DatabasebetaFinite Group Databasebeta| order | 32 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | Aut(G) = #2021 (order 128) · Inn(G) = C22 · Out(G) = #120 |
| holomorph | G ⋊ Aut(G) = #72219551 |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C2×C4 | C22 |
| commutator | C2 | C42 |
| socle, prime radical | C22 | C2×C4 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
20 conjugacy classes of subgroups.
| minimal degree | μ(G) = 12 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 2 generators on 12 points:
(5,6,7,10,8,9,11,12);(1,2,3,4)(6,9)(10,12)Polycyclic presentation:
2^5|a^2=c,b^2=d,c^2=e,b^a=be20 conjugacy classes — view the full character table.
| Frattini subgroup of | #1355, #1356, C2×#1355, … |
| Fitting subgroup of | #97×A5, #14514927, #14515003 |
| maximal subgroup of | #329, #330, #347, … |
| maximal normal subgroup of | #329, #330, #347, … |
| maximal quotient of | #322, #336, #347, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32, 4);