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Dic205 #11717954

order820
isoclinismisoclinic class of D205
automorphismAut(G) order 65600
homologyAbelianization (H1) = C4 · Schur multiplier (H2) = C1

Aliases

dicyclicDic205

Splittings

derived, HallC5×C41 ⋊ C4
SylowC41 ⋊ Dic5
SylowC5 ⋊ Dic41

Subgroups

subgroupG/H
center, Frattini, hypercenterC2D205
commutator, hypocenterC5×C41C4
Fitting, socle, prime radical, cosocleC2×C5×C41C2
LCSG ▷ C5×C41
UCSC1 ◁ C2

12 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 50
transitive degreeμt(G) = 820

A minimal faithful permutation representation — 2 generators on 50 points:

(1,2)(3,4)(5,6,7,8,9)(10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50);(1,3,2,4)(6,9)(7,8)(11,50)(12,49)(13,48)(14,47)(15,46)(16,45)(17,44)(18,43)(19,42)(20,41)(21,40)(22,39)(23,38)(24,37)(25,36)(26,35)(27,34)(28,33)(29,32)(30,31)

Polycyclic presentation:

2^2,5,41|a^2=b,c^a=c^4,d^a=d^40

Conjugacy classes & character table

208 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#13109982, #13109983, Dic5×D41, …
maximal normal subgroup of#13109982, #13109983, Dic5×D41, …
maximal quotient of#13109969, C2×Dic205, C3×Dic205, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(820, 5);