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C5⋊(D7⋊A4) #11719862

order840
isoclinismisoclinic class of C5⋊(D7⋊A4)
automorphism?
homologyAbelianization (H1) = C2×C3 · Schur multiplier (H2) = C2

Aliases

unambiguous semidirect productC5⋊(D7⋊A4)
unambiguous semidirect productD35⋊A4
unambiguous semidirect product(C22⋊ΓL(1,𝔽8))⋊D5

Splittings

Fitting, socle, derivedC22×C5×C7 ⋊ C2×C3
HallC22×D35 ⋊ C3
HallC5×C7 ⋊ C2×A4
SylowC7 ⋊ D5×A4
SylowC5 ⋊ D7⋊A4

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC22×C5×C7C2×C3
LCSG ▷ C22×C5×C7
UCSC1

48 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 140

Polycyclic presentation:

2,3,2^2,5,7|e^a=e^4,f^a=f^6,c^b=d,d^b=cd,f^b=f^2

Conjugacy classes & character table

36 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD5×D7⋊A4, C2×C5⋊(D7⋊A4), A4×C5⋊AGL(1,𝔽7)
maximal normal subgroup ofD5×D7⋊A4, C2×C5⋊(D7⋊A4), A4×C5⋊AGL(1,𝔽7)
maximal quotient ofC5⋊#66665, C5⋊(D7⋊SL(2,𝔽3)), (C22⋊ΓL(1,𝔽8))⋊Dic5, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(840, 164);