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C2×C3×C5⋊(ΓL(1,𝔽8)⋊D29) #49141504

order36540
automorphism?
homologyAbelianization (H1) = C22×C32 · Schur multiplier (H2) = C2×C3

Aliases

direct productC2×C3×C5⋊(ΓL(1,𝔽8)⋊D29)
unambiguous semidirect product(C2×C5)⋊(C3×ΓL(1,𝔽8)⋊D29)
unambiguous semidirect product(C2×ΓL(1,𝔽8))⋊(C3×D145)
unambiguous semidirect product(C2×C29)⋊(C3×C5⋊AGL(1,𝔽7))
unambiguous semidirect product(C2×D1015)⋊C32
unambiguous semidirect product(C2×C3×D1015)⋊C3

Splittings

Fitting, socleC2×C3×C5×C7×C29 ⋊ C2×C3
derived, HallC5×C7×C29 ⋊ C22×C32
HallC3×C5×ΓL(1,𝔽8)×C29 ⋊ C22
HallC2×D1015 ⋊ C32
HallC3×ΓL(1,𝔽8)×C29 ⋊ C2×D5
HallC3×C5×ΓL(1,𝔽8) ⋊ C2×D29
HallC7×C29 ⋊ C2×C32×D5
HallC5×C29 ⋊ C2×C3×AGL(1,𝔽7)
HallC3×ΓL(1,𝔽8) ⋊ C2×D145
HallC5×C7 ⋊ C2×C32×D29
SylowC29 ⋊ C2×C3×C5⋊AGL(1,𝔽7)
SylowC7 ⋊ C2×C32×D145
SylowC5 ⋊ C2×C3×ΓL(1,𝔽8)⋊D29

Subgroups

subgroupG/H
center, hypercenterC2×C3C5⋊(ΓL(1,𝔽8)⋊D29)
commutator, cosocle, hypocenterC5×C7×C29C22×C32
FrattiniC1G
Fitting, socleC2×C3×C5×C7×C29C2×C3
LCSG ▷ C5×C7×C29
UCSC1 ◁ C2×C3

240 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 82 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) = 46
transitive degreeμt(G) = 6090

A minimal faithful permutation representation — 2 generators on 46 points:

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

Conjugacy classes & character table

2202 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(36540, 346);