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C2×C3×C5×GL(2,𝔽3) #13089049

order1440
isoclinismisoclinic class of #226
automorphism?
homologyAbelianization (H1) = C22×C3×C5 · Schur multiplier (H2) = C2

Aliases

direct productC2×C3×C5×GL(2,𝔽3)

Splittings

FittingC2×C3×C5×Q8 ⋊ D3
derivedSL(2,𝔽3) ⋊ C22×C3×C5

Subgroups

subgroupG/H
center, socle, hypercenterC22×C3×C5S4
commutator, cosocle, hypocenterSL(2,𝔽3)C22×C3×C5
FrattiniC2C2×C3×C5×S4
FittingC2×C3×C5×Q8D3
LCSG ▷ SL(2,𝔽3)
UCSC1 ◁ C22×C3×C5

218 conjugacy classes of subgroups; only the 48 normal subgroups are shown.

normal subgroups only

This group has 218 conjugacy classes of subgroups — too many to draw. Only its 48 normal subgroups are shown.

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

Full subgroup data →

Presentation

transitive degreeμt(G) = 240

Conjugacy classes & character table

240 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×C3×C5×AGL(2,𝔽3), C2×C3×D5×GL(2,𝔽3), C5×AGL(1,𝔽7)×GL(2,𝔽3), …
maximal normal subgroup ofC2×C3×D5×GL(2,𝔽3), C22×C3×C5×GL(2,𝔽3), C3×C4×C5×GL(2,𝔽3)
maximal quotient ofC2×C3×C5×AGL(2,𝔽3), C5×AGL(1,𝔽7)×GL(2,𝔽3), C22×C3×C5×GL(2,𝔽3), …
automorphism group ofC32×C31
outer automorphism group ofC32×C31

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1440, 5366);