Finite Group Databasebeta

← Prev (#14514697) Next (#14514699) →

C22×AGL(1,𝔽5)×S4 #14514698

order1920
isoclinismisoclinic class of AGL(1,𝔽5)×S4
automorphism?
homologyAbelianization (H1) = C23×C4 · Schur multiplier (H2) = C27

Aliases

direct productC22×AGL(1,𝔽5)×S4
unambiguous semidirect productD5⋊(C2×C4×S4)
unambiguous semidirect product(A4⋊AGL(1,𝔽5))⋊C23
unambiguous semidirect product(A4×AGL(1,𝔽5))⋊C23
unambiguous semidirect product(D5×S4)⋊(C2×C4)

Splittings

Fitting, socleC24×C5 ⋊ C4×D3
derivedC5×A4 ⋊ C23×C4
SylowC5 ⋊ C22×C4×S4

Subgroups

subgroupG/H
center, hypercenterC22AGL(1,𝔽5)×S4
commutator, hypocenterC5×A4C23×C4
FrattiniC1G
Fitting, socleC24×C5C4×D3
prime radicalC22×D5×S4C2
cosocleD5×A4C24
LCSG ▷ C5×A4
UCSC1 ◁ C22

2582 conjugacy classes of subgroups; only the 198 normal subgroups are shown.

normal subgroups only

This group has 2582 conjugacy classes of subgroups — too many to draw. Only its 198 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) = 60

Conjugacy classes & character table

100 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×AGL(1,𝔽5)×#5935, C2×AGL(1,𝔽5)×GL(2,ℤ/4), A4×AGL(1,𝔽5)×S4, …
maximal normal subgroup ofC2×AGL(1,𝔽5)×GL(2,ℤ/4), A4×AGL(1,𝔽5)×S4, D4×AGL(1,𝔽5)×S4
maximal quotient ofC2×AGL(1,𝔽5)×GL(2,ℤ/4), D4×AGL(1,𝔽5)×S4, C22×#72214019, …
automorphism group of#4094, D5×#41, C3×Q8×D5, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1920, 240396);