Finite Group DatabasebetaFinite Group Databasebeta| order | 136 |
| isoclinism | isoclinic class of D17 |
| automorphism | Aut(G) = C22×AGL(1,𝔽17) (order 1088) · Inn(G) = D17 · Out(G) = C22×C8 |
| homology | Abelianization (H1) = C8 · Schur multiplier (H2) = C1 |
No aliases for this group.
| derived, Sylow | C17 ⋊ C8 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C4 | D17 |
| commutator, hypocenter | C17 | C8 |
| Fitting, cosocle | C4×C17 | C2 |
| socle, prime radical | C2×C17 | C4 |
| LCS | G ▷ C17 | |
| UCS | C1 ◁ C4 | |
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)(4,5)(6,7)(8,9)(10,11)(12,13)(14,15)(16,17)(18,19,20,22,21,23,24,25);(1,2,4,6,8,10,12,14,16,17,15,13,11,9,7,5,3)Polycyclic presentation:
2^3,17|a^2=b,b^2=c,d^a=d^1640 conjugacy classes — view the full character table.
| maximal subgroup of | #63216, C8×D17, #63218, … |
| maximal normal subgroup of | #63216, C8×D17, #63218, … |
| maximal quotient of | #63214, C2×#3622, C3×#3622, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(136, 1);