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Cayley diagram

C22×C3⋊AGL(1,𝔽7) #92754

order504
isoclinismisoclinic class of C3⋊AGL(1,𝔽7)
automorphism?
homologyAbelianization (H1) = C23×C3 · Schur multiplier (H2) = C23

Aliases

direct productC22×C3⋊AGL(1,𝔽7)

Splittings

Fitting, socleC22×C3×C7 ⋊ C2×C3
derivedC3×C7 ⋊ C23×C3
HallC3×ΓL(1,𝔽8) ⋊ C23
SylowC7 ⋊ C22×C3×D3

Subgroups

subgroupG/H
center, hypercenterC22C3⋊AGL(1,𝔽7)
commutator, cosocle, hypocenterC3×C7C23×C3
FrattiniC1G
Fitting, socleC22×C3×C7C2×C3
LCSG ▷ C3×C7
UCSC1 ◁ C22

138 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 14
transitive degreeμt(G) = 84

A minimal faithful permutation representation — 3 generators on 14 points:

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

Conjugacy classes & character table

48 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#66575⋊C3, #66631⋊C3, C2×(C3⋊AGL(1,𝔽7))⋊C4, …
maximal normal subgroup of#66575⋊C3, #66631⋊C3, C2×(C3⋊AGL(1,𝔽7))⋊C4, …
maximal quotient ofC2×Dic42⋊C3, C2×C4×C3⋊AGL(1,𝔽7), C2×C4⋊(C3⋊AGL(1,𝔽7)), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(504, 180);