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C₂²⋊Q₈×AGL(1,𝔽₇) #13057364

order	1344
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product	C₂²⋊Q₈×AGL(1,𝔽₇)
unambiguous semidirect product	D₇⋊(C₃×C₂²⋊Q₈)
unambiguous semidirect product	(D₇×C₂²⋊Q₈)⋊C₃

Splittings

Fitting	C₇×C₂²⋊Q₈ ⋊ C₂×C₃
Hall	D₇×C₂²⋊Q₈ ⋊ C₃
Hall	ΓL(1,𝔽₈) ⋊ C₂×C₂²⋊Q₈
Sylow	C₇ ⋊ C₂×C₃×C₂²⋊Q₈

Subgroups

	subgroup	G/H
center, Frattini	C₂²	C₂³×AGL(1,𝔽₇)
commutator, socle, cosocle	C₂²×C₇	C₂⁴×C₃
Fitting	C₇×C₂²⋊Q₈	C₂×C₃
prime radical	C₂³×AGL(1,𝔽₇)	C₂²
hypercenter	C₂²⋊Q₈	AGL(1,𝔽₇)
hypocenter	C₇	C₂×C₃×C₂²⋊Q₈
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₂²⋊Q₈

644 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 112

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	AGL(1,𝔽₇)×#525, AGL(1,𝔽₇)×#546, AGL(1,𝔽₇)×#538, …
maximal normal subgroup of	AGL(1,𝔽₇)×#525, AGL(1,𝔽₇)×#546, AGL(1,𝔽₇)×#538, …
maximal quotient of	AGL(1,𝔽₇)×#385

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 6653);