Finite Group Databasebeta

← Prev (#11929534) Next (#11929536) →

C3⋊(C3⋊(C7⋊AGL(1,𝔽5))) #11929535

order1260
isoclinismisoclinic class of C3⋊(C3⋊(C7⋊AGL(1,𝔽5)))
automorphism?
homologyAbelianization (H1) = C4 · Schur multiplier (H2) = C3

Aliases

unambiguous semidirect productC3⋊(C3⋊(C7⋊AGL(1,𝔽5)))
unambiguous semidirect productC7⋊(C3⋊ΓL(1,𝔽16))

Splittings

Fitting, socle, derived, HallC32×C5×C7 ⋊ C4
HallC32×C7 ⋊ AGL(1,𝔽5)
HallC32×C5 ⋊ Dic7
HallC5×C7 ⋊ C3⋊Dic3
SylowC32 ⋊ C7⋊AGL(1,𝔽5)
SylowC7 ⋊ C3⋊ΓL(1,𝔽16)
SylowC5 ⋊ C3⋊Dic21

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, hypocenterC32×C5×C7C4
prime radical, cosocleC32×C7×D5C2
LCSG ▷ C32×C5×C7
UCSC1

72 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 315

Polycyclic presentation:

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

Conjugacy classes & character table

129 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC11×C3⋊(C3⋊(C7⋊AGL(1,𝔽5))), C3⋊(C3⋊(C7⋊(C11⋊AGL(1,𝔽5)))), C13×C3⋊(C3⋊(C7⋊AGL(1,𝔽5))), …
maximal normal subgroup ofC11×C3⋊(C3⋊(C7⋊AGL(1,𝔽5))), C13×C3⋊(C3⋊(C7⋊AGL(1,𝔽5))), C17×C3⋊(C3⋊(C7⋊AGL(1,𝔽5))), …
maximal quotient ofC11×C3⋊(C3⋊(C7⋊AGL(1,𝔽5))), C3⋊(C3⋊(C7⋊(C11⋊AGL(1,𝔽5)))), C13×C3⋊(C3⋊(C7⋊AGL(1,𝔽5))), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1260, 87);