Finite Group DatabasebetaFinite Group Databasebeta| order | 416 |
| isoclinism | isoclinic class of D13 |
| automorphism | ? |
| homology | Abelianization (H1) = C32 · Schur multiplier (H2) = C1 |
No aliases for this group.
| derived, Sylow | C13 ⋊ C32 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C16 | D13 |
| commutator, hypocenter | C13 | C32 |
| Fitting, cosocle | C13×C16 | C2 |
| socle, prime radical | C2×C13 | C16 |
| LCS | G ▷ C13 | |
| UCS | C1 ◁ C16 | |
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)(4,5)(6,7)(8,9)(10,11)(12,13)(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);(1,2,4,6,8,10,12,13,11,9,7,5,3)Polycyclic presentation:
2^5,13|a^2=b,b^2=c,c^2=d,d^2=e,f^a=f^12128 conjugacy classes — view the full character table.
| maximal subgroup of | #11718037, D13×C32, #11718039, … |
| maximal normal subgroup of | #11718037, D13×C32, #11718039, … |
| maximal quotient of | #11718035, C2×#88034, C3×#88034, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(416, 1);