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 | D4×C13 | C4 |
| prime radical | #6129 | C2 |
| cosocle | Dic13 | C23 |
| hypercenter | D4 | #261 |
| hypocenter | C13 | #131 |
| LCS | G ▷ C2×C13 ▷ C13 | |
| UCS | C1 ◁ C2 ◁ D4 | |
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)(16,22)(19,25)(23,28)(26,29);(14,16)(15,19)(17,23)(18,22)(20,26)(21,25)(24,28)(27,29);(2,3,4,6)(5,8,9,13)(7,11,12,10)(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^8,c^b=ce,f^d=f^1235 conjugacy classes — view the full character table.
| maximal subgroup of | #11719103, #11719104, #11719107, … |
| maximal normal subgroup of | #11719103, #11719104, #11719107, … |
| maximal quotient of | #11719062, #11719063, #11719064, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(416, 205);