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C7×C9⋊AGL(1,𝔽5) #11929469

order1260
isoclinismisoclinic class of C9⋊AGL(1,𝔽5)
automorphism?
homologyAbelianization (H1) = C4×C7 · Schur multiplier (H2) = C1

Aliases

direct productC7×C9⋊AGL(1,𝔽5)
unambiguous semidirect productC9⋊(C7×AGL(1,𝔽5))

Splittings

Fitting, HallC5×C7×C9 ⋊ C4
derived, HallC5×C9 ⋊ C4×C7
HallC7×C9 ⋊ AGL(1,𝔽5)
HallC5×C7 ⋊ Dic9
SylowC9 ⋊ C7×AGL(1,𝔽5)
SylowC5 ⋊ C7×Dic9

Subgroups

subgroupG/H
center, hypercenterC7C9⋊AGL(1,𝔽5)
commutator, hypocenterC5×C9C4×C7
FrattiniC3C7×ΓL(1,𝔽16)
FittingC5×C7×C9C4
socleC3×C5×C7Dic3
prime radicalC3×C7×D5D3
cosocleC9×D5C2×C7
LCSG ▷ C5×C9
UCSC1 ◁ C7

36 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 2 generators on 21 points:

(1,2,3,4,5,6,7)(9,10,11,12)(14,16)(15,19)(17,18)(20,21);(8,9,12,10,11)(13,21,16,19,18,17,15,14,20)

Conjugacy classes & character table

147 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC7×C11×C9⋊AGL(1,𝔽5), C7×C9⋊(C11⋊AGL(1,𝔽5)), C7×C13×C9⋊AGL(1,𝔽5), …
maximal normal subgroup ofC7×C11×C9⋊AGL(1,𝔽5), C7×C13×C9⋊AGL(1,𝔽5), C7×C17×C9⋊AGL(1,𝔽5), …
maximal quotient ofC7×C11×C9⋊AGL(1,𝔽5), C7×C9⋊(C11⋊AGL(1,𝔽5)), C7×C13×C9⋊AGL(1,𝔽5), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1260, 21);