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AGL(1,𝔽7)×C43 #14268589

order1806
isoclinismisoclinic class of AGL(1,𝔽7)
automorphism?
homologyAbelianization (H1) = C2×C3×C43 · Schur multiplier (H2) = C1

Aliases

direct productAGL(1,𝔽7)×C43
unambiguous semidirect productD7⋊(C3×C43)
unambiguous semidirect productΓL(1,𝔽8)⋊(C2×C43)

Splittings

Fitting, socle, HallC7×C43 ⋊ C2×C3
derived, SylowC7 ⋊ C2×C3×C43
HallΓL(1,𝔽8)×C43 ⋊ C2
HallD7×C43 ⋊ C3
HallΓL(1,𝔽8) ⋊ C2×C43
HallD7 ⋊ C3×C43

Subgroups

subgroupG/H
center, hypercenterC43AGL(1,𝔽7)
commutator, cosocle, hypocenterC7C2×C3×C43
FrattiniC1G
Fitting, socleC7×C43C2×C3
LCSG ▷ C7
UCSC1 ◁ C43

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 50
transitive degreeμt(G) = 301

A minimal faithful permutation representation — 2 generators on 50 points:

(2,5,6,3,4,7)(8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50);(1,2,4,7,6,5,3)

Conjugacy classes & character table

301 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup ofAGL(1,𝔽7)×D43, C2×AGL(1,𝔽7)×C43, C43×D49⋊C3, …
maximal normal subgroup ofAGL(1,𝔽7)×D43, C2×AGL(1,𝔽7)×C43, AGL(1,𝔽7)×C43⋊C7, …
maximal quotient ofC43×ΓL(1,𝔽8)⋊C4, C2×AGL(1,𝔽7)×C43, C43×D49⋊C3, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1806, 13);