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C7⋊(C52⋊C3) #10587126

order525
isoclinismisoclinic class of C7⋊(C52⋊C3)
automorphism?
homologyAbelianization (H1) = C3 · Schur multiplier (H2) = C5

Aliases

unambiguous semidirect productC7⋊(C52⋊C3)
unambiguous semidirect productC52⋊ΓL(1,𝔽8)

Splittings

Fitting, socle, derived, HallC52×C7 ⋊ C3
SylowC52 ⋊ ΓL(1,𝔽8)
SylowC7 ⋊ C52⋊C3

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC52×C7C3
LCSG ▷ C52×C7
UCSC1

12 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 105

Polycyclic presentation:

3,5^2,7|b^a=b^3c^2,c^a=bc,d^a=d^2

Conjugacy classes & character table

61 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD7⋊(C52⋊C3), (C5⋊D5)⋊ΓL(1,𝔽8), #11760386, …
maximal normal subgroup ofD7⋊(C52⋊C3), (C5⋊D5)⋊ΓL(1,𝔽8), #11760386, …
maximal quotient ofC2×C7⋊(C52⋊C3), C7⋊(C52⋊C9), C3×C7⋊(C52⋊C3), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(525, 5);