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D₇×Dic₉ #92590

order	504
isoclinism	isoclinic class of D₇×D₉
automorphism	?
homology	Abelianization (H₁) = C₂×C₄ · Schur multiplier (H₂) = C₂

Aliases

direct product	D₇×Dic₉
unambiguous semidirect product	(C₉×D₇)⋊C₄

Splittings

derived, Hall	C₇×C₉ ⋊ C₂×C₄
Sylow	C₉ ⋊ C₄×D₇
Sylow	C₇ ⋊ C₂×Dic₉

Subgroups

	subgroup	G/H
center, hypercenter	C₂	D₇×D₉
commutator, hypocenter	C₇×C₉	C₂×C₄
Frattini	C₂×C₃	D₃×D₇
Fitting, cosocle	C₂×C₇×C₉	C₂²
socle	C₂×C₃×C₇	C₂×D₃
prime radical	C₂×C₃×D₇	D₃
LCS	G ▷ C₇×C₉
UCS	C₁ ◁ C₂

48 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 252

A minimal faithful permutation representation — 2 generators on 20 points:

(2,4)(3,7)(5,6)(8,9)(10,11,13,15,16,14,12)(17,18,19,20);(1,2,5,7,9,8,3,6,4)(11,12)(13,14)(15,16)

Conjugacy classes & character table

60 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₇×Dic₁₈, #11758980, #11758981, …
maximal normal subgroup of	D₇×Dic₁₈, #11758980, #11758981, …
maximal quotient of	D₇×C₉⋊C₈, #11758861, Dic₇×Dic₉, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(504, 16);