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Cayley diagram

ΓL(1,𝔽8)2 #89207

order441
isoclinismisoclinic class of ΓL(1,𝔽8)2
automorphism?
homologyAbelianization (H1) = C32 · Schur multiplier (H2) = C3

Aliases

direct productΓL(1,𝔽8)2
unambiguous semidirect product(C7×ΓL(1,𝔽8))⋊C3
unambiguous semidirect product#3861⋊C3
unambiguous semidirect product#3862⋊C3

Splittings

Fitting, socle, derived, SylowC72 ⋊ C32

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC72C32
LCSG ▷ C72
UCSC1

28 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 14
transitive degreeμt(G) = 21

A minimal faithful permutation representation — 2 generators on 14 points:

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

Conjugacy classes & character table

25 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#64418⋊C3, ΓL(1,𝔽8)×AGL(1,𝔽7), ΓL(1,𝔽8)⋊AGL(1,𝔽7), …
maximal normal subgroup of#64418⋊C3, ΓL(1,𝔽8)×AGL(1,𝔽7), ΓL(1,𝔽8)⋊AGL(1,𝔽7), …
maximal quotient ofC2×ΓL(1,𝔽8)2, ΓL(1,𝔽8)×C7⋊C9, #3862⋊C9, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(441, 9);