Finite Group Databasebeta

← Prev (#49141462) Next (#49141464) →

C5×(C3⋊AGL(1,𝔽7))⋊D29 #49141463

order36540
automorphism?
homologyAbelianization (H1) = C22×C3×C5 · Schur multiplier (H2) = C2

Aliases

direct productC5×(C3⋊AGL(1,𝔽7))⋊D29
unambiguous semidirect product(C3⋊AGL(1,𝔽7))⋊(C5×D29)
unambiguous semidirect productD87⋊(C5×AGL(1,𝔽7))
unambiguous semidirect product(ΓL(1,𝔽8)⋊D29)⋊(C5×D3)
unambiguous semidirect product(D21⋊D29)⋊(C3×C5)
unambiguous semidirect product(C5×D21⋊D29)⋊C3

Splittings

Fitting, socleC3×C5×C7×C29 ⋊ C22×C3
derivedC3×C7×C29 ⋊ C22×C3×C5
HallC3×C5×ΓL(1,𝔽8)×C29 ⋊ C22
HallC3×ΓL(1,𝔽8)×C29 ⋊ C22×C5
HallC5×C7×C29 ⋊ C2×C3×D3
HallC3×C5×ΓL(1,𝔽8) ⋊ C2×D29
HallC7×C29 ⋊ C2×C3×C5×D3
HallC5×C29 ⋊ D3×AGL(1,𝔽7)
HallC3×ΓL(1,𝔽8) ⋊ C2×C5×D29
HallC5×C7 ⋊ C3×D3×D29
SylowC29 ⋊ C5×D3×AGL(1,𝔽7)
SylowC7 ⋊ C3×C5×D3×D29

Subgroups

subgroupG/H
center, hypercenterC5(C3⋊AGL(1,𝔽7))⋊D29
commutator, cosocle, hypocenterC3×C7×C29C22×C3×C5
FrattiniC1G
Fitting, socleC3×C5×C7×C29C22×C3
LCSG ▷ C3×C7×C29
UCSC1 ◁ C5

176 conjugacy classes of subgroups; only the 50 normal subgroups are shown.

normal subgroups only

This group has 176 conjugacy classes of subgroups — too many to draw. Only its 50 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degreeμ(G) = 44
transitive degreeμt(G) = 3045

A minimal faithful permutation representation — 2 generators on 44 points:

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

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(36540, 305);