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Cayley diagram

C22×C3×ΓL(1,𝔽8) #6999

order252
isoclinismisoclinic class of ΓL(1,𝔽8)
automorphismAut(G) = D32×AGL(1,𝔽7) (order 1512) · Inn(G) = ΓL(1,𝔽8) · Out(G) = C2×D32
homologyAbelianization (H1) = C22×C32 · Schur multiplier (H2) = C2×C3

Aliases

direct productC22×C3×ΓL(1,𝔽8)

Splittings

Fitting, socleC22×C3×C7 ⋊ C3
derived, SylowC7 ⋊ C22×C32
HallC22×C7 ⋊ C32

Subgroups

subgroupG/H
center, hypercenterC22×C3ΓL(1,𝔽8)
commutator, cosocle, hypocenterC7C22×C32
FrattiniC1G
Fitting, socleC22×C3×C7C3
LCSG ▷ C7
UCSC1 ◁ C22×C3

60 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:

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

Conjugacy classes & character table

60 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×C3×ΓL(1,𝔽8)⋊C4, C3×(C22⋊D7)⋊C3, C2×Dic3×ΓL(1,𝔽8), …
maximal normal subgroup ofC2×C3×ΓL(1,𝔽8)⋊C4, C3×(C22⋊D7)⋊C3, C2×Dic3×ΓL(1,𝔽8), …
maximal quotient ofC2×C3×C4×ΓL(1,𝔽8), C3×D4×ΓL(1,𝔽8), C3×Q8×ΓL(1,𝔽8), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(252, 38);