Finite Group DatabasebetaFinite Group Databasebeta| order | 416 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| Fitting | C2×Q8×C13 ⋊ C2 |
| Sylow | C13 ⋊ #124 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C22×D13 |
| commutator, socle, cosocle | C22×C13 | C23 |
| Fitting | C2×Q8×C13 | C2 |
| prime radical | C2×D52 | C2 |
| hypercenter | C2×Q8 | D13 |
| hypocenter | C13 | #124 |
| LCS | G ▷ C22×C13 ▷ C13 | |
| UCS | C1 ◁ C22 ◁ C2×Q8 | |
76 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)(18,20,21,25)(19,23,22,24);(18,19,21,22)(20,24,25,23);(2,3)(4,5)(6,7)(8,9)(10,11)(12,13)(14,15)(16,17)(19,22)(23,24)Polycyclic presentation:
2^5,13|b^2=e,c^2=e,b^a=be,c^a=cd,f^a=f^12,c^b=ce74 conjugacy classes — view the full character table.
| maximal subgroup of | #11718069, #11718299, #11718300, … |
| maximal normal subgroup of | #11718069, #11718299, #11718300, … |
| maximal quotient of | #11718638, #11718647, #11718649, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(416, 168);