Finite Group DatabasebetaFinite Group Databasebeta| order | 208 |
| isoclinism | isoclinic class of #6104 |
| automorphism | Aut(G) = C2×D4×AGL(1,𝔽13) (order 2496) · Inn(G) = C22⋊D13 · Out(G) = C2×C3×C4 |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Fitting | Q8×C13 ⋊ C2 |
| Sylow | C13 ⋊ SD16 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C22⋊D13 |
| commutator, cosocle | C4×C13 | C22 |
| Frattini | C4 | C2×D13 |
| Fitting | Q8×C13 | C2 |
| socle | C2×C13 | D4 |
| prime radical | D52 | C2 |
| hypercenter | Q8 | D13 |
| hypocenter | C13 | SD16 |
| LCS | G ▷ C4×C13 ▷ C2×C13 ▷ C13 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ Q8 | |
20 conjugacy classes of subgroups.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 104 |
A minimal faithful permutation representation — 2 generators on 21 points:
(1,2,4,6,8,10,12,13,11,9,7,5,3)(14,15,17,19)(16,21,20,18);(2,3)(4,5)(6,7)(8,9)(10,11)(12,13)(15,18)(16,20)(19,21)Polycyclic presentation:
2^4,13|b^2=d,c^2=d,b^a=bc,c^a=cd,e^a=e^12,c^b=cd37 conjugacy classes — view the full character table.
| maximal subgroup of | SD16×D13, #88168, #88172, … |
| maximal normal subgroup of | SD16×D13, #88168, #88172, … |
| maximal quotient of | #88048, #88049, #88075, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(208, 17);