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ΓL(1,𝔽8)⋊#1039 #14523568

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

Aliases

unambiguous semidirect productΓL(1,𝔽8)⋊#1039

Splittings

derived, HallC52×C7 ⋊ C3×C4
HallC7⋊#1039 ⋊ C3
HallC52×ΓL(1,𝔽8) ⋊ C4
SylowC52 ⋊ ΓL(1,𝔽8)⋊C4
HallΓL(1,𝔽8) ⋊ #1039
SylowC7 ⋊ C3×#1039

Subgroups

subgroupG/H
center, Frattini, hypercenterC2C5⋊(C5⋊AGL(1,𝔽7))
commutator, hypocenterC52×C7C3×C4
Fitting, socle, cosocleC2×C52×C7C2×C3
prime radicalC2×C52×ΓL(1,𝔽8)C2
LCSG ▷ C52×C7
UCSC1 ◁ C2

96 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 700

Polycyclic presentation:

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

Conjugacy classes & character table

134 conjugacy classes — view the full character table.

Relationships

maximal subgroup of(C49⋊C3)⋊#1039, C7×ΓL(1,𝔽8)⋊#1039, #28413153, …
maximal normal subgroup ofC7×ΓL(1,𝔽8)⋊#1039, C11×ΓL(1,𝔽8)⋊#1039, C13×ΓL(1,𝔽8)⋊#1039, …
maximal quotient of(C49⋊C3)⋊#1039, C7×ΓL(1,𝔽8)⋊#1039, #28413153, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(2100, 48);