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C4×C41⋊C5 #11717951

order820
isoclinismisoclinic class of C41⋊C5
automorphism?
homologyAbelianization (H1) = C4×C5 · Schur multiplier (H2) = C1

Aliases

direct productC4×C41⋊C5
unambiguous semidirect product(C4×C41)⋊C5

Splittings

Fitting, HallC4×C41 ⋊ C5
derived, SylowC41 ⋊ C4×C5

Subgroups

subgroupG/H
center, hypercenterC4C41⋊C5
commutator, hypocenterC41C4×C5
FrattiniC2C2×C41⋊C5
FittingC4×C41C5
socle, cosocleC2×C41C2×C5
prime radicalC2×C41⋊C5C2
LCSG ▷ C41
UCSC1 ◁ C4

12 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 45
transitive degreeμt(G) = 164

A minimal faithful permutation representation — 2 generators on 45 points:

(1,2,3,4)(6,7,9,13,21)(8,11,17,28,39)(10,15,24,16,26)(12,19,27,38,25)(14,23,34,22,33)(18,29,40,44,37)(20,32,31,42,45)(30,41,35,43,36);(5,6,8,12,20,29,33,38,44,40,7,10,16,27,34,41,13,22,9,14,11,18,30,42,43,45,23,35,15,25,19,31,28,39,24,36,17,21,26,37,32)

Conjugacy classes & character table

52 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#66491⋊C5, C8×C41⋊C5, (C41⋊C5)⋊Q8, …
maximal normal subgroup of#66491⋊C5, C8×C41⋊C5, (C41⋊C5)⋊Q8, …
maximal quotient ofC8×C41⋊C5, C2×C4×C41⋊C5, Dic3×C41⋊C5, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(820, 2);