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#1039⋊ΓL(1,𝔽8) #14523573

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

Aliases

unambiguous semidirect product#1039⋊ΓL(1,𝔽8)

Splittings

derived, HallC52×C7 ⋊ C3×C4
HallC7×#1039 ⋊ C3
HallC7⋊(C52⋊C3) ⋊ C4
Hall#1039 ⋊ ΓL(1,𝔽8)
SylowC52 ⋊ C4×ΓL(1,𝔽8)
SylowC7 ⋊ #1039⋊C3

Subgroups

subgroupG/H
center, Frattini, hypercenterC2(C5⋊D5)⋊ΓL(1,𝔽8)
commutator, hypocenterC52×C7C3×C4
Fitting, socle, cosocleC2×C52×C7C2×C3
prime radicalC2×C7⋊(C52⋊C3)C2
LCSG ▷ C52×C7
UCSC1 ◁ C2

36 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 420

Polycyclic presentation:

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

Conjugacy classes & character table

76 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#1039⋊(C49⋊C3), C7×#1039⋊ΓL(1,𝔽8), #28413172, …
maximal normal subgroup ofC7×#1039⋊ΓL(1,𝔽8), C11×#1039⋊ΓL(1,𝔽8), C13×#1039⋊ΓL(1,𝔽8), …
maximal quotient of#1039⋊(C49⋊C3), C7×#1039⋊ΓL(1,𝔽8), #28413172, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(2100, 53);