Finite Group DatabasebetaFinite Group Databasebeta| order | 416 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | C13×C22⋊C4 ⋊ C2 |
| Sylow | C13 ⋊ #123 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C22×D13 |
| commutator, socle, cosocle | C22×C13 | C23 |
| Fitting | C13×C22⋊C4 | C2 |
| prime radical | C2×C22⋊D13 | C2 |
| hypercenter | C22⋊C4 | D13 |
| hypocenter | C13 | #123 |
| LCS | G ▷ C22×C13 ▷ C13 | |
| UCS | C1 ◁ C22 ◁ C22⋊C4 | |
78 conjugacy classes of subgroups.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 208 |
A minimal faithful permutation representation — 3 generators on 25 points:
(1,2,4,6,8,10,12,13,11,9,7,5,3)(15,17)(19,23)(22,25);(14,15)(16,17)(18,20,21,24)(19,22,23,25);(2,3)(4,5)(6,7)(8,9)(10,11)(12,13)(18,19)(20,22)(21,23)(24,25)Polycyclic presentation:
2^5,13|b^2=d,c^a=cd,f^a=f^12,c^b=ce74 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#88137, #11719195, #11719199, … |
| maximal normal subgroup of | C2×#88137, #11719195, #11719199, … |
| maximal quotient of | #11718313, #11718315, #11718318, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(416, 104);