Finite Group DatabasebetaFinite Group Databasebeta| order | 136 |
| isoclinism | isoclinic class of #595 |
| automorphism | Aut(G) = C2×AGL(1,𝔽17) (order 544) · Inn(G) = #595 · Out(G) = C2×C4 |
| homology | Abelianization (H1) = C8 · Schur multiplier (H2) = C1 |
No aliases for this group.
| derived, Sylow | C17 ⋊ C8 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | #595 |
| commutator, hypocenter | C17 | C8 |
| Fitting, socle, prime radical | C2×C17 | C4 |
| cosocle | Dic17 | C2 |
| LCS | G ▷ C17 | |
| UCS | C1 ◁ C2 | |
8 conjugacy classes of subgroups.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 136 |
A minimal faithful permutation representation — 2 generators on 25 points:
(2,3,4,6)(5,8,9,13)(7,12,10,11)(14,16,17,15)(18,19,20,22,21,23,24,25);(1,2,5,10,6,11,15,17,13,8,14,16,12,3,7,9,4)Polycyclic presentation:
2^3,17|a^2=b,b^2=c,d^a=d^13,d^b=d^1616 conjugacy classes — view the full character table.
| maximal subgroup of | #63241, #63242, #63243, … |
| maximal normal subgroup of | #63241, #63242, #63243, … |
| maximal quotient of | #63216, C2×#3624, C3×#3624, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(136, 3);