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ΓL(1,𝔽8)×#1035 #14523534

order2100
isoclinismisoclinic class of ΓL(1,𝔽8)×#1035
automorphism?
homologyAbelianization (H1) = C3×C4 · Schur multiplier (H2) = C1

Aliases

direct productΓL(1,𝔽8)×#1035
unambiguous semidirect product(C7×#1035)⋊C3

Splittings

Fitting, derived, HallC7×C25 ⋊ C3×C4
HallC7×#1035 ⋊ C3
HallΓL(1,𝔽8)×C25 ⋊ C4
SylowC25 ⋊ C4×ΓL(1,𝔽8)
SylowC7 ⋊ C3×#1035

Subgroups

subgroupG/H
center, hypercenterC1G
commutator, Fitting, hypocenterC7×C25C3×C4
FrattiniC5AGL(1,𝔽5)×ΓL(1,𝔽8)
socleC5×C7C3×AGL(1,𝔽5)
prime radicalΓL(1,𝔽8)×D25C2
cosocleC7×D25C2×C3
LCSG ▷ C7×C25
UCSC1

36 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 32
transitive degreeμt(G) = 175

A minimal faithful permutation representation — 2 generators on 32 points:

(2,3,4)(5,7,6)(9,11,12,17)(10,15,16,27)(13,21,22,26)(14,24,25,18)(19,29,30,32)(20,23,28,31);(1,2,3,5,4,6,7)(8,9,13,23,20,10,14,11,18,29,15,26,19,30,21,27,32,24,17,25,16,28,31,22,12)

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#1035×C49⋊C3, C7×ΓL(1,𝔽8)×#1035, #28413031, …
maximal normal subgroup ofC7×ΓL(1,𝔽8)×#1035, C11×ΓL(1,𝔽8)×#1035, C13×ΓL(1,𝔽8)×#1035, …
maximal quotient of#1035×C49⋊C3, C7×ΓL(1,𝔽8)×#1035, #28413031, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(2100, 14);