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C2×D245 #11758254

order980
isoclinismisoclinic class of D245
automorphismAut(G) order 82320
homologyAbelianization (H1) = C22 · Schur multiplier (H2) = C2

Aliases

dihedralD490
direct productC2×D245
unambiguous semidirect product(C2×C5)⋊D49
unambiguous semidirect product(C2×C49)⋊D5

Splittings

FittingC2×C5×C49 ⋊ C2
derived, HallC5×C49 ⋊ C22
SylowC49 ⋊ C2×D5
SylowC5 ⋊ C2×D49

Subgroups

subgroupG/H
center, hypercenterC2D245
commutator, cosocle, hypocenterC5×C49C22
FrattiniC7C2×D35
FittingC2×C5×C49C2
socleC2×C5×C7D7
LCSG ▷ C5×C49
UCSC1 ◁ C2

30 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 56
transitive degreeμt(G) = 490

A minimal faithful permutation representation — 2 generators on 56 points:

(1,2)(3,4,5,6,7)(8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56);(4,7)(5,6)(9,56)(10,55)(11,54)(12,53)(13,52)(14,51)(15,50)(16,49)(17,48)(18,47)(19,46)(20,45)(21,44)(22,43)(23,42)(24,41)(25,40)(26,39)(27,38)(28,37)(29,36)(30,35)(31,34)(32,33)

Conjugacy classes & character table

248 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#14519912, #14519913, #14519916, …
maximal normal subgroup of#14519912, #14519913, #14519916, …
maximal quotient ofDic490, C4×D245, D980, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(980, 10);