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D₇×(C₃⋊D₃)⋊C₃ #10625459

order	756
isoclinism	isoclinic class of D₇×(C₃⋊D₃)⋊C₃
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃ · Schur multiplier (H₂) = C₂×C₃

Aliases

D₇×(C₃⋊D₃)⋊C₃

Splittings

Fitting, Hall	C₇×H(3,𝔽₃) ⋊ C₂²
derived	C₃²×C₇ ⋊ C₂²×C₃
Sylow	H(3,𝔽₃) ⋊ C₂×D₇
Sylow	C₇ ⋊ C₂×(C₃⋊D₃)⋊C₃

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, cosocle, hypocenter	C₃²×C₇	C₂²×C₃
Frattini	C₃	C₃×D₃×D₇
Fitting	C₇×H(3,𝔽₃)	C₂²
socle	C₃×C₇	C₂×C₃×D₃
LCS	G ▷ C₃²×C₇
UCS	C₁

78 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 63

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₇×((C₃⋊D₃)⋊C₃)⋊C₂, C₂×D₇×(C₃⋊D₃)⋊C₃, D₇×PSU(3,𝔽₂)⋊C₃, …
maximal normal subgroup of	D₇×((C₃⋊D₃)⋊C₃)⋊C₂, C₂×D₇×(C₃⋊D₃)⋊C₃, AGL(1,𝔽₇)×(C₃⋊D₃)⋊C₃, …
maximal quotient of	#13094340, Dic₇×(C₃⋊D₃)⋊C₃, #13094342, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(756, 90);