Finite Group DatabasebetaFinite Group Databasebeta| order | 416 |
| isoclinism | isoclinic class of #88066 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Sylow | C13 ⋊ Q32 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #6104 |
| commutator, cosocle | C8×C13 | C22 |
| Frattini | C8 | C2×D13 |
| Fitting | C13×Q16 | C2 |
| socle, prime radical | C2×C13 | D8 |
| hypercenter | Q16 | D13 |
| hypocenter | C13 | Q32 |
| LCS | G ▷ C8×C13 ▷ C4×C13 ▷ C2×C13 ▷ C13 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 ◁ Q16 | |
24 conjugacy classes of subgroups.
| minimal degree | μ(G) = 45 |
| transitive degree | μt(G) = 416 |
A minimal faithful permutation representation — 2 generators on 45 points:
(1,2,4,6,8,10,12,13,11,9,7,5,3)(14,16,19,27)(15,20,23,37)(17,29,32,45)(18,34,30,26)(21,39,40,44)(22,41,33,36)(24,42,43,28)(25,38,35,31);(2,3)(4,5)(6,7)(8,9)(10,11)(12,13)(14,15,19,23)(16,24,27,43)(17,28,32,42)(18,33,30,22)(20,35,37,25)(21,38,40,31)(26,44,34,39)(29,41,45,36)Polycyclic presentation:
2^5,13|a^2=e,b^2=e,c^2=de,d^2=e,b^a=bc,c^a=cd,d^a=de,f^a=f^12,c^b=cd,d^b=de53 conjugacy classes — view the full character table.
| maximal subgroup of | #11718576, #11718577, D13×Q32, … |
| maximal normal subgroup of | #11718576, #11718577, D13×Q32, … |
| maximal quotient of | #11718081, #11718084, #11718156, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(416, 36);