Finite Group DatabasebetaFinite Group Databasebeta| order | 208 |
| automorphism | Aut(G) = C2×S4×AGL(1,𝔽13) (order 7488) · Inn(G) = C22×D13 · Out(G) = C22×C3×D3 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | Q8×C13 ⋊ C2 |
| Sylow | C13 ⋊ #41 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C22×D13 |
| commutator, socle, cosocle | C2×C13 | C23 |
| Fitting | Q8×C13 | C2 |
| hypercenter | Q8 | D13 |
| hypocenter | C13 | #41 |
| LCS | G ▷ C2×C13 ▷ C13 | |
| UCS | C1 ◁ C2 ◁ Q8 | |
40 conjugacy classes of subgroups.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 104 |
A minimal faithful permutation representation — 3 generators on 21 points:
(1,2,4,6,8,10,12,13,11,9,7,5,3)(14,16,17,20)(15,18,19,21);(14,15,17,19)(16,21,20,18);(2,3)(4,5)(6,7)(8,9)(10,11)(12,13)(16,20)(18,21)Polycyclic presentation:
2^4,13|b^2=d,c^2=d,c^a=cd,e^a=e^12,c^b=cd40 conjugacy classes — view the full character table.
| maximal subgroup of | #88118, #88168, #88170, … |
| maximal normal subgroup of | #88118, #88168, #88170, … |
| maximal quotient of | #88144, #88146, #88147, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(208, 42);