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ΓL(1,𝔽8)×#1039 #14523567

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

Aliases

direct productΓL(1,𝔽8)×#1039

Splittings

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

Subgroups

subgroupG/H
center, Frattini, hypercenterC2ΓL(1,𝔽8)×C5⋊D5
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

minimal degreeμ(G) = 21
transitive degreeμt(G) = 700

A minimal faithful permutation representation — 3 generators on 21 points:

(2,3,4)(5,7,6)(8,9,10,11)(13,14)(15,16)(18,19)(20,21);(1,2,3,5,4,6,7)(17,18,20,21,19);(12,13,15,16,14)

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#1039×C49⋊C3, C7×ΓL(1,𝔽8)×#1039, #28413155, …
maximal normal subgroup ofC7×ΓL(1,𝔽8)×#1039, C11×ΓL(1,𝔽8)×#1039, C13×ΓL(1,𝔽8)×#1039, …
maximal quotient of#1039×C49⋊C3, C7×ΓL(1,𝔽8)×#1039, #28413155, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(2100, 47);