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 | C2 | C22×#261 |
| commutator, socle | C2×C13 | C22×C4 |
| Fitting | Q8×C13 | C4 |
| prime radical | #6131 | C2 |
| cosocle | Dic13 | C23 |
| hypercenter | Q8 | #261 |
| hypocenter | C13 | #131 |
| LCS | G ▷ C2×C13 ▷ C13 | |
| UCS | C1 ◁ C2 ◁ Q8 | |
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:
(1,2,5,8,12,6,10,11,3,7,13,9,4)(14,17,19,18)(15,21,23,22)(16,24,25,26)(20,27,28,29);(14,16,19,25)(15,20,23,28)(17,26,18,24)(21,29,22,27);(2,3,4,6)(5,8,9,13)(7,11,12,10)(14,15,18,22,19,23,17,21)(16,20,24,27,25,28,26,29)Polycyclic presentation:
2^5,13|a^2=d,b^2=e,c^2=e,d^2=e,f^a=f^8,c^b=ce,f^d=f^1235 conjugacy classes — view the full character table.
| maximal subgroup of | #11719107, #11719108, #11719111, … |
| maximal normal subgroup of | #11719107, #11719108, #11719111, … |
| maximal quotient of | #11719072, #11719073, #11719075, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(416, 207);