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C7⋊S5 #11719834

order840
isoclinismisoclinic class of C7⋊S5
automorphism?
homologyAbelianization (H1) = C2 · Schur multiplier (H2) = C2

Aliases

unambiguous semidirect productC7⋊S5
unambiguous semidirect productA5⋊D7

Splittings

Fitting, SylowC7 ⋊ S5
socle, derivedC7×A5 ⋊ C2

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, socle, cosocle, hypocenterC7×A5C2
FittingC7S5
LCSG ▷ C7×A5
UCSC1

38 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 12
transitive degreeμt(G) = 35

A minimal faithful permutation representation — 2 generators on 12 points:

(2,7)(3,6)(4,5)(8,12,11,9);(1,5)(2,4)(6,7)(8,12,9)(10,11)

Conjugacy classes & character table

22 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD7×S5, C2×C7⋊S5
maximal normal subgroup ofD7×S5, C2×C7⋊S5
maximal quotient ofC7⋊(A5⋊C4), C7⋊#6728, C7⋊#6729, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(840, 136);