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C₇×C₉×C₃⋊C₈ #13094171

order	1512
isoclinism	isoclinic class of D₃
automorphism	?
homology	Abelianization (H₁) = C₇×C₈×C₉ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₇×C₉×C₃⋊C₈
unambiguous semidirect product	C₃⋊(C₇×C₈×C₉)

Splittings

derived	C₃ ⋊ C₇×C₈×C₉
Hall	C₃×C₇×C₉ ⋊ C₈
Sylow	C₃×C₉ ⋊ C₇×C₈

Subgroups

	subgroup	G/H
center, hypercenter	C₄×C₇×C₉	D₃
commutator, hypocenter	C₃	C₇×C₈×C₉
Frattini	C₃×C₄	C₃×D₃×C₇
Fitting	C₃×C₄×C₇×C₉	C₂
socle, prime radical	C₂×C₃²×C₇	C₃×C₄
cosocle	C₃²×C₄	C₂×C₃×C₇
LCS	G ▷ C₃
UCS	C₁ ◁ C₄×C₇×C₉

60 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 27
transitive degree	μ_t(G) = 504

Conjugacy classes & character table

756 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂×C₇×C₉×C₃⋊C₈, D₃×C₇×C₈×C₉
maximal normal subgroup of	C₂×C₇×C₉×C₃⋊C₈, D₃×C₇×C₈×C₉
maximal quotient of	C₂×C₇×C₉×C₃⋊C₈

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1512, 68);