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Cayley diagram

#41×ΓL(1,𝔽8) #66698

order336
isoclinismisoclinic class of D4×ΓL(1,𝔽8)
automorphism?
homologyAbelianization (H1) = C23×C3 · Schur multiplier (H2) = C22

Aliases

direct product#41×ΓL(1,𝔽8)

Splittings

Fitting, HallC7×#41 ⋊ C3
SylowC7 ⋊ C3×#41

Subgroups

subgroupG/H
centerC4C22×ΓL(1,𝔽8)
commutator, socle, cosocleC2×C7C23×C3
FrattiniC2C23×ΓL(1,𝔽8)
FittingC7×#41C3
hypercenter#41ΓL(1,𝔽8)
hypocenterC7C3×#41
LCSG ▷ C2×C7 ▷ C7
UCSC1 ◁ C4 ◁ #41

80 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 15
transitive degreeμt(G) = 56

A minimal faithful permutation representation — 3 generators on 15 points:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#6411⋊C3, ΓL(1,𝔽8)×#104, #10620784, …
maximal normal subgroup of#6411⋊C3, ΓL(1,𝔽8)×#104, #10620784, …
maximal quotient ofΓL(1,𝔽8)×#117, C4×D4×ΓL(1,𝔽8), C4×Q8×ΓL(1,𝔽8), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(336, 167);