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Cayley diagram

Dic3×ΓL(1,𝔽8) #6978

order252
isoclinismisoclinic class of D3×ΓL(1,𝔽8)
automorphismAut(G) = C2×D3×AGL(1,𝔽7) (order 504) · Inn(G) = D3×ΓL(1,𝔽8) · Out(G) = C22
homologyAbelianization (H1) = C3×C4 · Schur multiplier (H2) = C1

Aliases

direct productDic3×ΓL(1,𝔽8)
unambiguous semidirect productC3⋊(C4×ΓL(1,𝔽8))
unambiguous semidirect product(C7×Dic3)⋊C3

Splittings

derivedC3×C7 ⋊ C3×C4
HallC3×ΓL(1,𝔽8) ⋊ C4
SylowC7 ⋊ C3×Dic3

Subgroups

subgroupG/H
center, Frattini, hypercenterC2D3×ΓL(1,𝔽8)
commutator, hypocenterC3×C7C3×C4
Fitting, socle, cosocleC2×C3×C7C2×C3
prime radicalC2×C3×ΓL(1,𝔽8)C2
LCSG ▷ C3×C7
UCSC1 ◁ C2

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

Conjugacy classes & character table

30 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofDic3×AGL(1,𝔽7), (C3⋊AGL(1,𝔽7))⋊C4, #4246⋊C3, …
maximal normal subgroup ofDic3×AGL(1,𝔽7), (C3⋊AGL(1,𝔽7))⋊C4, #4246⋊C3, …
maximal quotient ofΓL(1,𝔽8)×C3⋊C8, C2×Dic3×ΓL(1,𝔽8), #10625391, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(252, 17);