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C₄⋊SO(3,𝔽₇) #13062000

order	1344
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂²

Aliases

unambiguous semidirect product

C₄⋊SO(3,𝔽₇)

Splittings

C₄ ⋊ SO(3,𝔽₇)

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×SO(3,𝔽₇)
commutator, socle, cosocle	C₂×GL(3,𝔽₂)	C₂²
Fitting, hypercenter	C₄	SO(3,𝔽₇)
hypocenter	GL(3,𝔽₂)	D₄
LCS	G ▷ C₂×GL(3,𝔽₂) ▷ GL(3,𝔽₂)
UCS	C₁ ◁ C₂ ◁ C₄

122 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 32

Permutation generators:

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

Conjugacy classes & character table

24 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72244484
maximal normal subgroup of	#72244484

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 11289);