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ΓL(1,𝔽8)⋊#1044 #14523595

order2100
isoclinismisoclinic class of ΓL(1,𝔽8)⋊#1044
automorphism?
homologyAbelianization (H1) = C3×C4 · Schur multiplier (H2) = C5

Aliases

unambiguous semidirect productΓL(1,𝔽8)⋊#1044
unambiguous semidirect product(C7⋊#1044)⋊C3

Splittings

Fitting, socle, derived, HallC52×C7 ⋊ C3×C4
HallC7⋊#1044 ⋊ C3
HallC52×ΓL(1,𝔽8) ⋊ C4
SylowC52 ⋊ ΓL(1,𝔽8)⋊C4
HallΓL(1,𝔽8) ⋊ #1044
SylowC7 ⋊ C3×#1044

Subgroups

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

64 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 70

Polycyclic presentation:

2,3,2,5^2,7|a^2=c,d^a=d^2,e^a=e^3,f^a=f^6,f^b=f^2,d^c=d^4,e^c=e^4

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

Relationships

maximal subgroup of(C49⋊C3)⋊#1044, C7×ΓL(1,𝔽8)⋊#1044, #28413266, …
maximal normal subgroup ofC7×ΓL(1,𝔽8)⋊#1044, C11×ΓL(1,𝔽8)⋊#1044, C13×ΓL(1,𝔽8)⋊#1044, …
maximal quotient of(C49⋊C3)⋊#1044, C7×ΓL(1,𝔽8)⋊#1044, #28413266, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(2100, 75);