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C2×D5605 #36375783

order22420
automorphism?
homologyAbelianization (H1) = C22 · Schur multiplier (H2) = C2

Aliases

dihedralD11210
direct productC2×D5605
unambiguous semidirect product(C2×C5)⋊D1121
unambiguous semidirect product(C2×C19)⋊D295
unambiguous semidirect product(C2×C59)⋊D95

Splittings

Fitting, socleC2×C5×C19×C59 ⋊ C2
derived, HallC5×C19×C59 ⋊ C22
HallC19×C59 ⋊ C2×D5
HallC5×C59 ⋊ C2×D19
HallC5×C19 ⋊ C2×D59
SylowC59 ⋊ C2×D95
SylowC19 ⋊ C2×D295
SylowC5 ⋊ C2×D1121

Subgroups

subgroupG/H
center, hypercenterC2D5605
commutator, cosocle, hypocenterC5×C19×C59C22
FrattiniC1G
Fitting, socleC2×C5×C19×C59C2
LCSG ▷ C5×C19×C59
UCSC1 ◁ C2

40 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 85
transitive degreeμt(G) = 11210

A minimal faithful permutation representation — 2 generators on 85 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)(27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85);(4,7)(5,6)(9,26)(10,25)(11,24)(12,23)(13,22)(14,21)(15,20)(16,19)(17,18)(28,85)(29,84)(30,83)(31,82)(32,81)(33,80)(34,79)(35,78)(36,77)(37,76)(38,75)(39,74)(40,73)(41,72)(42,71)(43,70)(44,69)(45,68)(46,67)(47,66)(48,65)(49,64)(50,63)(51,62)(52,61)(53,60)(54,59)(55,58)(56,57)

Conjugacy classes & character table

5608 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup ofD22420
maximal normal subgroup ofD22420
maximal quotient ofDic11210, D22420

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(22420, 27);