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C7⋊(C7⋊AGL(1,𝔽5)) #11758265

order980
isoclinismisoclinic class of C7⋊(C7⋊AGL(1,𝔽5))
automorphism?
homologyAbelianization (H1) = C4 · Schur multiplier (H2) = C7

Aliases

unambiguous semidirect productC7⋊(C7⋊AGL(1,𝔽5))

Splittings

Fitting, socle, derived, HallC5×C72 ⋊ C4
SylowC72 ⋊ AGL(1,𝔽5)
SylowC5 ⋊ C7⋊Dic7

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, hypocenterC5×C72C4
prime radical, cosocleC72×D5C2
LCSG ▷ C5×C72
UCSC1

60 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 245

Polycyclic presentation:

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

Conjugacy classes & character table

101 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofAGL(1,𝔽5)×C7⋊D7, D7×C7⋊AGL(1,𝔽5), C2×C7⋊(C7⋊AGL(1,𝔽5)), …
maximal normal subgroup ofAGL(1,𝔽5)×C7⋊D7, D7×C7⋊AGL(1,𝔽5), C2×C7⋊(C7⋊AGL(1,𝔽5)), …
maximal quotient ofC7⋊(C7⋊#170), C2×C7⋊(C7⋊AGL(1,𝔽5)), C3×C7⋊(C7⋊AGL(1,𝔽5)), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(980, 21);