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D₉×Dic₇ #92587

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

Aliases

direct product	D₉×Dic₇
unambiguous semidirect product	(C₇×D₉)⋊C₄

Splittings

derived, Hall	C₇×C₉ ⋊ C₂×C₄
Sylow	C₉ ⋊ C₂×Dic₇
Sylow	C₇ ⋊ C₄×D₉

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×C₇×D₉	C₂
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:

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

Conjugacy classes & character table

60 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₉×Dic₁₄, #11758976, #11758983, …
maximal normal subgroup of	D₉×Dic₁₄, #11758976, #11758983, …
maximal quotient of	D₉×C₇⋊C₈, #11758854, Dic₇×Dic₉, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(504, 13);