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Cayley diagram

C2×C41⋊C5 #88012

order410
isoclinismisoclinic class of C41⋊C5
automorphismAut(G) = AGL(1,𝔽41) (order 1640) · Inn(G) = C41⋊C5 · Out(G) = C8
homologyAbelianization (H1) = C2×C5 · Schur multiplier (H2) = C1

Aliases

direct productC2×C41⋊C5
unambiguous semidirect product(C2×C41)⋊C5

Splittings

Fitting, socle, HallC2×C41 ⋊ C5
derived, SylowC41 ⋊ C2×C5

Subgroups

subgroupG/H
center, hypercenterC2C41⋊C5
commutator, cosocle, hypocenterC41C2×C5
FrattiniC1G
Fitting, socleC2×C41C5
LCSG ▷ C41
UCSC1 ◁ C2

8 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 43
transitive degreeμt(G) = 82

A minimal faithful permutation representation — 2 generators on 43 points:

(1,2)(4,5,7,11,19)(6,9,15,26,37)(8,13,22,14,24)(10,17,25,36,23)(12,21,32,20,31)(16,27,38,42,35)(18,30,29,40,43)(28,39,33,41,34);(3,4,6,10,18,27,31,36,42,38,5,8,14,25,32,39,11,20,7,12,9,16,28,40,41,43,21,33,13,23,17,29,26,37,22,34,15,19,24,35,30)

Conjugacy classes & character table

26 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofDic41⋊C5, C4×C41⋊C5, C2×D41⋊C5, …
maximal normal subgroup ofDic41⋊C5, C4×C41⋊C5, C2×D41⋊C5, …
maximal quotient ofC4×C41⋊C5, C22×C41⋊C5, D3×C41⋊C5, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(410, 2);