Finite Group DatabasebetaFinite Group Databasebeta| order | 136 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | Aut(G) = C16×S4 (order 384) · Inn(G) = C22 · Out(G) = D3×C16 |
| holomorph | G ⋊ Aut(G) = #5935×AGL(1,𝔽17) |
| homology | Abelianization (H1) = C22×C17 · Schur multiplier (H2) = C1 |
| direct product | Q8×C17 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C2×C17 | C22 |
| commutator, Frattini, cosocle | C2 | C22×C17 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C17 ◁ G | |
12 conjugacy classes of subgroups.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 136 |
A minimal faithful permutation representation — 2 generators on 25 points:
(1,2,3,4)(5,8,7,6)(9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25);(1,5,3,7)(2,6,4,8)85 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(136, 11);