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C2×D205 #11717968

order820
isoclinismisoclinic class of D205
automorphismAut(G) order 65600
homologyAbelianization (H1) = C22 · Schur multiplier (H2) = C2

Aliases

dihedralD410
direct productC2×D205
unambiguous semidirect product(C2×C5)⋊D41
unambiguous semidirect product(C2×C41)⋊D5

Splittings

Fitting, socleC2×C5×C41 ⋊ C2
derived, HallC5×C41 ⋊ C22
SylowC41 ⋊ C2×D5
SylowC5 ⋊ C2×D41

Subgroups

subgroupG/H
center, hypercenterC2D205
commutator, cosocle, hypocenterC5×C41C22
FrattiniC1G
Fitting, socleC2×C5×C41C2
LCSG ▷ C5×C41
UCSC1 ◁ C2

20 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 48
transitive degreeμt(G) = 410

A minimal faithful permutation representation — 2 generators on 48 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);(4,7)(5,6)(9,48)(10,47)(11,46)(12,45)(13,44)(14,43)(15,42)(16,41)(17,40)(18,39)(19,38)(20,37)(21,36)(22,35)(23,34)(24,33)(25,32)(26,31)(27,30)(28,29)

Conjugacy classes & character table

208 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#13109988, #13109990, #13109991, …
maximal normal subgroup of#13109988, #13109990, #13109991, …
maximal quotient ofDic410, C4×D205, D820, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(820, 19);