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C7×(C31⋊C3)⋊AGL(1,𝔽5) #27189143

order13020
automorphism?
homologyAbelianization (H1) = C3×C4×C7 · Schur multiplier (H2) = C1

Aliases

direct productC7×(C31⋊C3)⋊AGL(1,𝔽5)
unambiguous semidirect product(C31⋊C3)⋊(C7×AGL(1,𝔽5))
unambiguous semidirect product(C31⋊AGL(1,𝔽5))⋊(C3×C7)

Splittings

Fitting, socle, HallC5×C7×C31 ⋊ C3×C4
derived, HallC5×C31 ⋊ C3×C4×C7
HallC7×C31⋊AGL(1,𝔽5) ⋊ C3
HallC5×C7×C31⋊C3 ⋊ C4
HallC7×C31⋊C3 ⋊ AGL(1,𝔽5)
HallC31⋊AGL(1,𝔽5) ⋊ C3×C7
HallC5×C31⋊C3 ⋊ C4×C7
HallC7×C31 ⋊ C3×AGL(1,𝔽5)
HallC31⋊C3 ⋊ C7×AGL(1,𝔽5)
HallC5×C7 ⋊ (C31⋊C3)⋊C4
SylowC31 ⋊ C3×C7×AGL(1,𝔽5)
SylowC5 ⋊ C7×(C31⋊C3)⋊C4

Subgroups

subgroupG/H
center, hypercenterC7(C31⋊C3)⋊AGL(1,𝔽5)
commutator, hypocenterC5×C31C3×C4×C7
FrattiniC1G
Fitting, socleC5×C7×C31C3×C4
prime radicalC7×D5×C31⋊C3C2
cosocleD5×C31C2×C3×C7
LCSG ▷ C5×C31
UCSC1 ◁ C7

48 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 43
transitive degreeμt(G) = 1085

A minimal faithful permutation representation — 2 generators on 43 points:

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

Conjugacy classes & character table

245 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13020, 110);