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AGL(1,𝔽5)×C7⋊C9 #11929461

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

Aliases

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

Splittings

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

Subgroups

subgroupG/H
center, Frattini, hypercenterC3AGL(1,𝔽5)×ΓL(1,𝔽8)
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

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

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

Conjugacy classes & character table

75 conjugacy classes — view the full character table.

Relationships

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1260, 13);