Finite Group DatabasebetaFinite Group Databasebeta| order | 416 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Fitting | C13×C4⋊C4 ⋊ C2 |
| Sylow | C13 ⋊ #126 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C22×D13 |
| commutator, socle, cosocle | C22×C13 | C23 |
| Fitting | C13×C4⋊C4 | C2 |
| prime radical | C22×D13 | C22 |
| hypercenter | C4⋊C4 | D13 |
| hypocenter | C13 | #126 |
| LCS | G ▷ C22×C13 ▷ C13 | |
| UCS | C1 ◁ C22 ◁ C4⋊C4 | |
60 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)(15,19)(18,21)(22,24,25,28)(23,27,26,29);(14,16,17,20)(15,18,19,21)(23,26)(24,28);(2,3)(4,5)(6,7)(8,9)(10,11)(12,13)(14,15)(16,18)(17,19)(20,21)(22,23)(24,27)(25,26)(28,29)Polycyclic presentation:
2^5,13|b^2=d,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 | C2×#88152, #11719218, #11719219, … |
| maximal normal subgroup of | C2×#88152, #11719218, #11719219, … |
| maximal quotient of | #11718312, #11718315, #11718321, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(416, 119);