Finite Group DatabasebetaFinite Group Databasebeta| order | 416 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | C13×SD16 ⋊ C2 |
| Sylow | C13 ⋊ #135 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | D4×D13 |
| commutator, cosocle | C4×C13 | C23 |
| Frattini | C4 | C22×D13 |
| Fitting | C13×SD16 | C2 |
| socle | C2×C13 | C2×D4 |
| hypercenter | SD16 | D13 |
| hypocenter | C13 | #135 |
| LCS | G ▷ C4×C13 ▷ C2×C13 ▷ C13 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ SD16 | |
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)(14,16,18,24)(15,20,19,22)(17,26,25,23)(21,28,27,29);(15,19)(16,23)(17,25)(20,28)(22,29)(24,26);(2,3)(4,5)(6,7)(8,9)(10,11)(12,13)(14,15)(16,22)(17,21)(18,19)(20,24)(23,28)(25,27)(26,29)Polycyclic presentation:
2^5,13|c^2=e,d^2=e,b^a=be,c^a=ce,f^a=f^12,c^b=cd,d^b=de,d^c=de56 conjugacy classes — view the full character table.
| maximal subgroup of | #11719107, #11719108, C2×#88170, … |
| maximal normal subgroup of | #11719107, #11719108, C2×#88170, … |
| maximal quotient of | #11718417, #11718424, #11718429, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(416, 137);