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#1043⋊ΓL(1,𝔽8) #14523609

order2100
isoclinismisoclinic class of #1043⋊ΓL(1,𝔽8)
automorphism?
homologyAbelianization (H1) = C3×C4 · Schur multiplier (H2) = C1

Aliases

unambiguous semidirect product#1043⋊ΓL(1,𝔽8)

Splittings

Fitting, socle, derived, HallC52×C7 ⋊ C3×C4
HallC7×#1043 ⋊ C3
HallC7⋊(C52⋊C3) ⋊ C4
Hall#1043 ⋊ ΓL(1,𝔽8)
SylowC52 ⋊ C4×ΓL(1,𝔽8)
SylowC7 ⋊ #1043⋊C3

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, hypocenterC52×C7C3×C4
prime radical(C5⋊D5)⋊ΓL(1,𝔽8)C2
cosocleC7×C5⋊D5C2×C3
LCSG ▷ C52×C7
UCSC1

36 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 105

Polycyclic presentation:

2,3,2,5^2,7|a^2=c,d^a=d^3,e^a=e^3,d^b=de^2,e^b=de^3,f^b=f^2,d^c=d^4,e^c=e^4

Conjugacy classes & character table

34 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#1043⋊(C49⋊C3), C7×#1043⋊ΓL(1,𝔽8), #28413359, …
maximal normal subgroup ofC7×#1043⋊ΓL(1,𝔽8), C11×#1043⋊ΓL(1,𝔽8), C13×#1043⋊ΓL(1,𝔽8), …
maximal quotient of#1043⋊(C49⋊C3), C7×#1043⋊ΓL(1,𝔽8), #28413359, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(2100, 89);