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Cayley diagram

C22⋊D41 #66498

order328
isoclinismisoclinic class of Dic82
automorphism?
homologyAbelianization (H1) = C22 · Schur multiplier (H2) = C2

Aliases

unambiguous semidirect productC22⋊D41

Splittings

FittingC22×C41 ⋊ C2
SylowC41 ⋊ D4

Subgroups

subgroupG/H
center, FrattiniC2C2×D41
commutator, socle, cosocleC2×C41C22
FittingC22×C41C2
hypercenterC22D41
hypocenterC41D4
LCSG ▷ C2×C41 ▷ C41
UCSC1 ◁ C2 ◁ C22

16 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:

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

Polycyclic presentation:

2^3,41|b^a=bc,d^a=d^40

Conjugacy classes & character table

85 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#10620306, D4×D41, #10620308, …
maximal normal subgroup of#10620306, D4×D41, #10620308, …
maximal quotient of#10620279, #10620281, #10620282, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(328, 8);