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D₆₃⋊C₄ #92588

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

Aliases

unambiguous semidirect product

D₆₃⋊C₄

Splittings

derived, Hall	C₇×C₉ ⋊ C₂×C₄
Sylow	C₉ ⋊ C₄×D₇
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×D₆₃	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:

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

Polycyclic presentation:

2^3,3,7,3|a^2=c,b^2=c,d^3=f^2,e^a=e^6,d^b=d^2f,f^b=f^2

Conjugacy classes & character table

60 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#11758977, D₆₃⋊Q₈, #11758984, …
maximal normal subgroup of	#11758977, D₆₃⋊Q₈, #11758984, …
maximal quotient of	D₆₃⋊C₈, #11758855, Dic₇×Dic₉, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(504, 14);