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C7×AGL(1,𝔽7)×D23 #27214136

order13524
automorphism?
homologyAbelianization (H1) = C22×C3×C7 · Schur multiplier (H2) = C2

Aliases

direct productC7×AGL(1,𝔽7)×D23
unambiguous semidirect productD7⋊(C3×C7×D23)
unambiguous semidirect product(D7×D23)⋊(C3×C7)
unambiguous semidirect product(ΓL(1,𝔽8)×D23)⋊(C2×C7)
unambiguous semidirect product(C23×AGL(1,𝔽7))⋊(C2×C7)
unambiguous semidirect product(ΓL(1,𝔽8)⋊D23)⋊(C2×C7)

Splittings

Fitting, socle, HallC72×C23 ⋊ C22×C3
derivedC7×C23 ⋊ C22×C3×C7
HallC7×D7×D23 ⋊ C3
HallC7×ΓL(1,𝔽8)×C23 ⋊ C22
HallC7×ΓL(1,𝔽8) ⋊ C2×D23
SylowC72 ⋊ C2×C3×D23
SylowC23 ⋊ C2×C7×AGL(1,𝔽7)

Subgroups

subgroupG/H
center, hypercenterC7AGL(1,𝔽7)×D23
commutator, cosocle, hypocenterC7×C23C22×C3×C7
FrattiniC1G
Fitting, socleC72×C23C22×C3
LCSG ▷ C7×C23
UCSC1 ◁ C7

84 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 37
transitive degreeμt(G) = 966

A minimal faithful permutation representation — 2 generators on 37 points:

(2,3)(4,5)(6,7)(8,9)(10,11)(12,13)(14,15)(16,17)(18,19)(20,21)(22,23)(24,25,26,27,28,29,30)(31,32,37)(34,35,36);(1,2,4,6,8,10,12,14,16,18,20,22,23,21,19,17,15,13,11,9,7,5,3)(32,33)(34,35)(36,37)

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13524, 70);