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C₇×C₉ #317

order	63
nilpotency class	1
isoclinism	isoclinic class of C₁
automorphism	Aut(G) = C₂²×C₃² (order 36) · Inn(G) = C₁ · Out(G) = C₂²×C₃²
holomorph	G ⋊ Aut(G) = AGL(1,𝔽₇)×AGL(1,ℤ/9)
homology	Abelianization (H₁) = G · Schur multiplier (H₂) = C₁

Aliases

cyclic	C₆₃
abelian	C₇×C₉

Subgroups

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₃	C₃×C₇
socle, prime radical	C₃×C₇	C₃
LCS	G ▷ C₁
UCS	C₁ ◁ G

6 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 16
transitive degree	μ_t(G) = 63

A minimal faithful permutation representation — 1 generator on 16 points:

(1,2,3,4,5,6,7,8,9)(10,11,12,13,14,15,16)

Conjugacy classes & character table

63 conjugacy classes — view the full character table.

Relationships

center of	D₃×C₇×C₉, C₇×C₂₇⋊C₃, C₇×#737, …
derived subgroup of	D₆₃, Dic₆₃, D₇×D₉, …
Frattini subgroup of	C₄₉⋊C₂₇, C₂₇×C₄₉, C₂×D₆₆₁₅, …
Fitting subgroup of	C₇×D₉, C₉×D₇, D₆₃, …
maximal subgroup of	C₇×D₉, C₉×D₇, D₆₃, …
maximal normal subgroup of	C₇×D₉, C₉×D₇, D₆₃, …
maximal quotient of	C₂×C₇×C₉, C₇×C₂₇, C₃×C₇×C₉, …
outer automorphism group of	D₁₂₇

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(63, 2);