Finite Group DatabasebetaFinite Group Databasebeta| order | 416 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C13 ⋊ #131 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C4 | C22×D13 |
| commutator, socle | C2×C13 | C22×C4 |
| Fitting, prime radical | C13×#41 | C2 |
| cosocle | C4×C13 | C23 |
| hypercenter | #41 | D13 |
| hypocenter | C13 | #131 |
| LCS | G ▷ C2×C13 ▷ C13 | |
| UCS | C1 ◁ C4 ◁ #41 | |
62 conjugacy classes of subgroups.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 208 |
A minimal faithful permutation representation — 3 generators on 29 points:
(1,2,4,6,8,10,12,13,11,9,7,5,3)(16,22)(19,25)(23,28)(26,29);(14,16)(15,19)(17,23)(18,22)(20,26)(21,25)(24,28)(27,29);(2,3)(4,5)(6,7)(8,9)(10,11)(12,13)(14,15,17,20,18,21,24,27)(16,19,23,26,22,25,28,29)Polycyclic presentation:
2^5,13|a^2=d,d^2=e,f^a=f^12,c^b=ce80 conjugacy classes — view the full character table.
| maximal subgroup of | #11718304, #11718484, #11718485, … |
| maximal normal subgroup of | #11718304, #11718484, #11718485, … |
| maximal quotient of | #11718658, #11718660, #11718661, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(416, 169);