Finite Group Databasebeta

← Prev (#92650) Next (#92652) →

C₃×C₄⋊AGL(1,𝔽₇) #92651

order	504
isoclinism	isoclinic class of ΓL(1,𝔽₈)⋊Q₈
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃² · Schur multiplier (H₂) = C₂×C₃

Aliases

direct product	C₃×C₄⋊AGL(1,𝔽₇)
unambiguous semidirect product	C₄⋊(C₃×AGL(1,𝔽₇))
unambiguous semidirect product	D₂₈⋊C₃²
unambiguous semidirect product	(C₃×D₂₈)⋊C₃

Splittings

Fitting	C₃×C₄×C₇ ⋊ C₂×C₃
Hall	C₃×ΓL(1,𝔽₈) ⋊ D₄
Hall	D₂₈ ⋊ C₃²
Sylow	C₇ ⋊ C₃²×D₄

Subgroups

	subgroup	G/H
center	C₂×C₃	C₂×AGL(1,𝔽₇)
commutator, cosocle	C₂×C₇	C₂²×C₃²
Frattini	C₂	C₂×C₃×AGL(1,𝔽₇)
Fitting	C₃×C₄×C₇	C₂×C₃
socle	C₂×C₃×C₇	C₂²×C₃
hypercenter	C₃×C₄	AGL(1,𝔽₇)
hypocenter	C₇	C₃²×D₄
LCS	G ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₃ ◁ C₃×C₄

96 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 14
transitive degree	μ_t(G) = 84

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

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

Conjugacy classes & character table

57 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#66560⋊C₃, #66561⋊C₃, #66563⋊C₃, …
maximal normal subgroup of	#66560⋊C₃, #66561⋊C₃, #66563⋊C₃, …
maximal quotient of	C₃×#1132⋊C₃, C₃×D₅₆⋊C₃, C₃×Dic₂₈⋊C₃, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(504, 77);