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C5×Dic5×ΓL(1,𝔽8) #14523565

order2100
isoclinismisoclinic class of D5×ΓL(1,𝔽8)
automorphism?
homologyAbelianization (H1) = C3×C4×C5 · Schur multiplier (H2) = C1

Aliases

direct productC5×Dic5×ΓL(1,𝔽8)
unambiguous semidirect product(C7×Dic5)⋊(C3×C5)
unambiguous semidirect product(C5×C7×Dic5)⋊C3

Splittings

derivedC5×C7 ⋊ C3×C4×C5
HallC5×C7×Dic5 ⋊ C3
HallC52×ΓL(1,𝔽8) ⋊ C4
HallC52×C7 ⋊ C3×C4
SylowC52 ⋊ C4×ΓL(1,𝔽8)
SylowC7 ⋊ C3×C5×Dic5

Subgroups

subgroupG/H
center, hypercenterC2×C5D5×ΓL(1,𝔽8)
commutator, hypocenterC5×C7C3×C4×C5
FrattiniC2C5×D5×ΓL(1,𝔽8)
Fitting, socleC2×C52×C7C2×C3
prime radicalC2×C52×ΓL(1,𝔽8)C2
cosocleC2×C5×C7C2×C3×C5
LCSG ▷ C5×C7
UCSC1 ◁ C2×C5

64 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 140

Conjugacy classes & character table

200 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC5×Dic5×C49⋊C3, C5×C7×Dic5×ΓL(1,𝔽8), C5×#15668736, …
maximal normal subgroup ofC5×C7×Dic5×ΓL(1,𝔽8), C5×C11×Dic5×ΓL(1,𝔽8), C5×C13×Dic5×ΓL(1,𝔽8), …
maximal quotient ofC5×Dic5×C49⋊C3, C5×C7×Dic5×ΓL(1,𝔽8), C5×#15668736, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(2100, 45);