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C₂×D₃×C₇⋊C₉ #10625448

order	756
isoclinism	isoclinic class of D₃×ΓL(1,𝔽₈)
automorphism	?
homology	Abelianization (H₁) = C₂²×C₉ · Schur multiplier (H₂) = C₂

Aliases

direct product	C₂×D₃×C₇⋊C₉
unambiguous semidirect product	C₃⋊(C₂²×C₇⋊C₉)
unambiguous semidirect product	(C₂×C₃)⋊(C₂×C₇⋊C₉)
unambiguous semidirect product	(C₂×D₃×C₇)⋊C₉

Splittings

derived	C₃×C₇ ⋊ C₂²×C₉
Hall	C₃×C₇⋊C₉ ⋊ C₂²
Sylow	C₇ ⋊ C₂×D₃×C₉

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₃	D₃×ΓL(1,𝔽₈)
commutator, hypocenter	C₃×C₇	C₂²×C₉
Frattini	C₃	C₂×D₃×ΓL(1,𝔽₈)
Fitting, socle	C₂×C₃²×C₇	C₂×C₃
prime radical	C₂×C₃×D₃×C₇	C₃
cosocle	C₃²×C₇	C₂²×C₃
LCS	G ▷ C₃×C₇
UCS	C₁ ◁ C₂×C₃

68 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 252

Conjugacy classes & character table

90 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×Dic₇⋊C₉, #4245⋊C₉, #4247⋊C₉, …
maximal normal subgroup of	D₃×Dic₇⋊C₉, #4245⋊C₉, #4247⋊C₉, …
maximal quotient of	Dic₆×C₇⋊C₉, C₄×D₃×C₇⋊C₉, D₁₂×C₇⋊C₉, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(756, 79);