Finite Group DatabasebetaFinite Group Databasebeta| order | 416 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | C2×D4×C13 ⋊ C2 |
| Sylow | C13 ⋊ H(5,𝔽2) |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C23×D13 |
| commutator, socle, cosocle | C2×C13 | C24 |
| Fitting | C2×D4×C13 | C2 |
| hypercenter | C2×D4 | D13 |
| hypocenter | C13 | H(5,𝔽2) |
| LCS | G ▷ C2×C13 ▷ C13 | |
| UCS | C1 ◁ C2 ◁ C2×D4 | |
166 conjugacy classes of subgroups.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 104 |
A minimal faithful permutation representation — 4 generators on 21 points:
(1,2,4,6,8,10,12,13,11,9,7,5,3)(15,19)(18,21);(16,20)(18,21);(14,16)(15,18)(17,20)(19,21);(2,3)(4,5)(6,7)(8,9)(10,11)(12,13)(14,15)(16,18)(17,19)(20,21)Polycyclic presentation:
2^5,13|d^a=de,f^a=f^12,c^b=ce77 conjugacy classes — view the full character table.
| maximal subgroup of | #11718402, #11718403, #11718407, … |
| maximal normal subgroup of | #11718402, #11718403, #11718407, … |
| maximal quotient of | #11719188, #11719191, #11719195, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(416, 218);