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(C7⋊C9)⋊AGL(1,𝔽5) #11929462

order1260
isoclinismisoclinic class of ΓL(1,𝔽8)⋊AGL(1,𝔽5)
automorphism?
homologyAbelianization (H1) = C4×C9 · Schur multiplier (H2) = C1

Aliases

unambiguous semidirect product(C7⋊C9)⋊AGL(1,𝔽5)
unambiguous semidirect product(C7⋊AGL(1,𝔽5))⋊C9

Splittings

derived, HallC5×C7 ⋊ C4×C9
HallC5×C7⋊C9 ⋊ C4
HallC7⋊AGL(1,𝔽5) ⋊ C9
HallC7⋊C9 ⋊ AGL(1,𝔽5)
SylowC7 ⋊ C9×AGL(1,𝔽5)
SylowC5 ⋊ Dic7⋊C9

Subgroups

subgroupG/H
center, Frattini, hypercenterC3ΓL(1,𝔽8)⋊AGL(1,𝔽5)
commutator, hypocenterC5×C7C4×C9
Fitting, socleC3×C5×C7C3×C4
prime radical, cosocleC3×C7×D5C2×C3
LCSG ▷ C5×C7
UCSC1 ◁ C3

36 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 21
transitive degreeμt(G) = 315

Polycyclic presentation:

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

Conjugacy classes & character table

57 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC11×(C7⋊C9)⋊AGL(1,𝔽5), C11⋊((C7⋊C9)⋊AGL(1,𝔽5)), C13×(C7⋊C9)⋊AGL(1,𝔽5), …
maximal normal subgroup ofC11×(C7⋊C9)⋊AGL(1,𝔽5), C13×(C7⋊C9)⋊AGL(1,𝔽5), C17×(C7⋊C9)⋊AGL(1,𝔽5), …
maximal quotient ofC11×(C7⋊C9)⋊AGL(1,𝔽5), C11⋊((C7⋊C9)⋊AGL(1,𝔽5)), C13×(C7⋊C9)⋊AGL(1,𝔽5), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1260, 14);