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ΓL(1,𝔽8)×#1044 #14523596

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

Aliases

direct 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 ⋊ C4×ΓL(1,𝔽8)
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

minimal degreeμ(G) = 17
transitive degreeμt(G) = 70

A minimal faithful permutation representation — 2 generators on 17 points:

(2,3,4)(5,7,6)(8,9)(10,13,11,15)(12,16,17,14);(1,2,3,5,4,6,7)(9,15,17,12,13)

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#1044×C49⋊C3, C7×ΓL(1,𝔽8)×#1044, #28413268, …
maximal normal subgroup ofC7×ΓL(1,𝔽8)×#1044, C11×ΓL(1,𝔽8)×#1044, C13×ΓL(1,𝔽8)×#1044, …
maximal quotient of#1044×C49⋊C3, C7×ΓL(1,𝔽8)×#1044, #28413268, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(2100, 76);