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C5×SL(2,𝔽7) #13132377

order1680
isoclinismisoclinic class of SL(2,𝔽7)
automorphism?
homologyAbelianization (H1) = C5 · Schur multiplier (H2) = C1

Aliases

direct productC5×SL(2,𝔽7)

Subgroups

subgroupG/H
center, Fitting, socle, hypercenterC2×C5GL(3,𝔽2)
commutator, hypocenterSL(2,𝔽7)C5
Frattini, cosocleC2C5×GL(3,𝔽2)
LCSG ▷ SL(2,𝔽7)
UCSC1 ◁ C2×C5

38 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 2 generators on 21 points:

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

Conjugacy classes & character table

55 conjugacy classes — view the full character table.

Relationships

derived subgroup ofDic5×SL(2,𝔽7)
maximal subgroup ofD5×SL(2,𝔽7), #72209750, C2×C5×SL(2,𝔽7)
maximal normal subgroup ofD5×SL(2,𝔽7), #72209750, C2×C5×SL(2,𝔽7)
maximal quotient ofC2×C5×SL(2,𝔽7)

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1680, 397);