Finite Group Databasebeta

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C2×C3×C7×ΓL(1,𝔽8)×C25 #36347764

order22050
automorphism?
homologyAbelianization (H1) = C2×C32×C7×C25 · Schur multiplier (H2) = C3

Aliases

direct productC2×C3×C7×ΓL(1,𝔽8)×C25
unambiguous semidirect product(C2×C7)⋊(C32×C7×C25)
unambiguous semidirect product(C2×C3×C7)⋊(C3×C7×C25)
unambiguous semidirect product(C2×C7×C25)⋊(C32×C7)
unambiguous semidirect product(C2×C3×C7×C25)⋊(C3×C7)

Splittings

FittingC2×C3×C72×C25 ⋊ C3
derivedC7 ⋊ C2×C32×C7×C25
HallC2×C72×C25 ⋊ C32
HallC72×C25 ⋊ C2×C32
HallC2×C72 ⋊ C32×C25
SylowC72 ⋊ C2×C32×C25

Subgroups

subgroupG/H
center, hypercenterC2×C3×C7×C25ΓL(1,𝔽8)
commutator, hypocenterC7C2×C32×C7×C25
FrattiniC5C2×C3×C5×C7×ΓL(1,𝔽8)
FittingC2×C3×C72×C25C3
socleC2×C3×C5×C72C3×C5
prime radicalC2×C3×C5×C7×ΓL(1,𝔽8)C5
cosocleC5×C7C2×C32×C5×C7
LCSG ▷ C7
UCSC1 ◁ C2×C3×C7×C25

168 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 168 conjugacy classes of subgroups — too many to draw. Only its 96 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) = 3150

Conjugacy classes & character table

5250 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(22050, 201);