Finite Group Databasebeta

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C5×#33551 #11956271

order1280
nilpotency class4
isoclinismisoclinic class of #33551
automorphism?
homologyAbelianization (H1) = C24×C5 · Schur multiplier (H2) = C26

Aliases

direct productC5×#33551

Subgroups

subgroupG/H
center, socleC2×C5C2×B(4,𝔽2)
commutator, Frattini, cosocleC2×D4C24×C5
LCSG ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C5 ◁ C23×C5 ◁ C22×C5×D4 ◁ G

1326 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 21
transitive degreeμt(G) = 80

A minimal faithful permutation representation — 4 generators on 21 points:

(1,2,3,4,5)(6,7)(8,14)(9,12)(10,13)(11,16)(15,20)(17,19)(18,21);(6,8)(7,11)(9,16)(10,15)(12,14)(13,18)(17,21)(19,20);(9,17)(16,21);(8,15)(11,18)(14,20)(16,21)

Conjugacy classes & character table

170 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of#72242889, AGL(1,𝔽5)×#33551, #72243102
maximal subgroup ofC5×#11715753, #72243102, C5×#511790
maximal normal subgroup ofD5×#33551, #72243102, C5×#511790

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 26539);