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C5×ΓL(1,𝔽8)⋊Dic5 #14523566

order2100
isoclinismisoclinic class of C5⋊AGL(1,𝔽7)
automorphism?
homologyAbelianization (H1) = C3×C4×C5 · Schur multiplier (H2) = C1

Aliases

direct productC5×ΓL(1,𝔽8)⋊Dic5
unambiguous semidirect productΓL(1,𝔽8)⋊(C5×Dic5)
unambiguous semidirect productDic35⋊(C3×C5)
unambiguous semidirect product(C5×Dic35)⋊C3

Splittings

derivedC5×C7 ⋊ C3×C4×C5
HallC5×Dic35 ⋊ C3
HallC52×ΓL(1,𝔽8) ⋊ C4
HallC52×C7 ⋊ C3×C4
SylowC52 ⋊ ΓL(1,𝔽8)⋊C4
HallΓL(1,𝔽8) ⋊ C5×Dic5
SylowC7 ⋊ C3×C5×Dic5

Subgroups

subgroupG/H
center, hypercenterC2×C5C5⋊AGL(1,𝔽7)
commutator, hypocenterC5×C7C3×C4×C5
FrattiniC2C5×C5⋊AGL(1,𝔽7)
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

170 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC5×(C49⋊C3)⋊Dic5, C5×C7×ΓL(1,𝔽8)⋊Dic5, C5×#15668734, …
maximal normal subgroup ofC5×C7×ΓL(1,𝔽8)⋊Dic5, C5×C11×ΓL(1,𝔽8)⋊Dic5, C5×C13×ΓL(1,𝔽8)⋊Dic5, …
maximal quotient ofC5×(C49⋊C3)⋊Dic5, C5×C7×ΓL(1,𝔽8)⋊Dic5, C5×#15668734, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(2100, 46);