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Cayley diagram

C3×C22⋊ΓL(1,𝔽8) #7001

order252
isoclinismisoclinic class of C22⋊ΓL(1,𝔽8)
automorphismAut(G) = D3×A4×AGL(1,𝔽7) (order 3024) · Inn(G) = C22⋊ΓL(1,𝔽8) · Out(G) = C2×C3×D3
homologyAbelianization (H1) = C32 · Schur multiplier (H2) = C2×C3

Aliases

direct productC3×C22⋊ΓL(1,𝔽8)
unambiguous semidirect product(C22×C3)⋊ΓL(1,𝔽8)
unambiguous semidirect product(C3×C7)⋊A4

Splittings

Fitting, socleC22×C3×C7 ⋊ C3
derived, HallC22×C7 ⋊ C32
SylowC7 ⋊ C3×A4
SylowC22 ⋊ C3×ΓL(1,𝔽8)

Subgroups

subgroupG/H
center, hypercenterC3C22⋊ΓL(1,𝔽8)
commutator, cosocle, hypocenterC22×C7C32
FrattiniC1G
Fitting, socleC22×C3×C7C3
LCSG ▷ C22×C7
UCSC1 ◁ C3

28 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Conjugacy classes & character table

36 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC3×D7⋊A4, D3×C22⋊ΓL(1,𝔽8), C3⋊(D7⋊A4), …
maximal normal subgroup ofC3×D7⋊A4, D3×C22⋊ΓL(1,𝔽8), C3⋊(D7⋊A4), …
maximal quotient ofC3×C7⋊SL(2,𝔽3), C2×C3×C22⋊ΓL(1,𝔽8), C9×C22⋊ΓL(1,𝔽8), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(252, 40);