Finite Group DatabasebetaFinite Group Databasebeta| order | 416 |
| isoclinism | isoclinic class of #261 |
| automorphism | Aut(G) = C8×AGL(1,𝔽13) (order 1248) · Inn(G) = #261 · Out(G) = C3×C8 |
| homology | Abelianization (H1) = C32 · Schur multiplier (H2) = C1 |
No aliases for this group.
| derived, Sylow | C13 ⋊ C32 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C8 | #261 |
| commutator, hypocenter | C13 | C32 |
| Fitting | C8×C13 | C4 |
| socle, prime radical | C2×C13 | C16 |
| cosocle | #6090 | C2 |
| LCS | G ▷ C13 | |
| UCS | C1 ◁ C8 | |
12 conjugacy classes of subgroups.
| minimal degree | μ(G) = 45 |
| transitive degree | μt(G) = 416 |
A minimal faithful permutation representation — 2 generators on 45 points:
(2,3,4,6)(5,8,9,13)(7,11,12,10)(14,15,16,20,17,21,24,30,18,22,25,31,27,33,36,40,19,23,26,32,28,34,37,41,29,35,38,42,39,43,44,45);(1,2,5,8,12,6,10,11,3,7,13,9,4)Polycyclic presentation:
2^5,13|a^2=b,b^2=c,c^2=d,d^2=e,f^a=f^8,f^b=f^1256 conjugacy classes — view the full character table.
| maximal subgroup of | D13⋊C32, #11718214, C2×#88036, … |
| maximal normal subgroup of | D13⋊C32, #11718214, C2×#88036, … |
| maximal quotient of | #11718037, C2×#88036, C3×#88036, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(416, 3);