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C2×#41×S4 #11715769

order768
automorphism?
homologyAbelianization (H1) = C25 · Schur multiplier (H2) = C210

Aliases

direct productC2×#41×S4

Splittings

FittingC23×#41 ⋊ D3

Subgroups

subgroupG/H
centerC2×C4C22×S4
commutator, cosocleC2×A4C25
FrattiniC2C24×S4
FittingC23×#41D3
socleC24C23×D3
hypercenterC2×#41S4
hypocenterA4C22×#41
LCSG ▷ C2×A4 ▷ A4
UCSC1 ◁ C2×C4 ◁ C2×#41

6015 conjugacy classes of subgroups; only the 525 normal subgroups are shown.

normal subgroups only

This group has 6015 conjugacy classes of subgroups — too many to draw. Only its 525 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) = 48

Conjugacy classes & character table

100 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72228056, C2×S4×AGL(1,ℤ/8), #72240983, …
maximal normal subgroup of#72228056, C2×S4×AGL(1,ℤ/8), #72258602, …
maximal quotient of#72228056, C2×#72220637, #41×GL(2,ℤ/4), …
outer automorphism group ofC2×#504

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 1090150);