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C2×C7×C5⋊(D31⋊C3) #27189202

order13020
automorphism?
homologyAbelianization (H1) = C22×C3×C7 · Schur multiplier (H2) = C2

Aliases

direct productC2×C7×C5⋊(D31⋊C3)
unambiguous semidirect product(C2×C5)⋊(C7×D31⋊C3)
unambiguous semidirect product(C2×C31⋊C3)⋊(C7×D5)
unambiguous semidirect product(C2×D155)⋊(C3×C7)

Splittings

Fitting, socleC2×C5×C7×C31 ⋊ C2×C3
derived, HallC5×C31 ⋊ C22×C3×C7
HallC2×C7×D155 ⋊ C3
HallC5×C7×C31⋊C3 ⋊ C22
HallC5×C7×C31 ⋊ C22×C3
HallC7×C31⋊C3 ⋊ C2×D5
HallC2×D155 ⋊ C3×C7
HallC5×C31⋊C3 ⋊ C22×C7
HallC7×C31 ⋊ C2×C3×D5
HallC31⋊C3 ⋊ C2×C7×D5
HallC5×C7 ⋊ C2×D31⋊C3
SylowC31 ⋊ C2×C3×C7×D5
SylowC5 ⋊ C2×C7×D31⋊C3

Subgroups

subgroupG/H
center, hypercenterC2×C7C5⋊(D31⋊C3)
commutator, cosocle, hypocenterC5×C31C22×C3×C7
FrattiniC1G
Fitting, socleC2×C5×C7×C31C2×C3
LCSG ▷ C5×C31
UCSC1 ◁ C2×C7

80 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 45
transitive degreeμt(G) = 2170

A minimal faithful permutation representation — 2 generators on 45 points:

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

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13020, 169);