C₉×C₄₇ #88320

order	423
nilpotency class	1
isoclinism	isoclinic class of C₁
automorphism	Aut(G) = C₂²×C₃×C₂₃ (order 276) · Inn(G) = C₁ · Out(G) = C₂²×C₃×C₂₃
homology	Abelianization (H₁) = G · Schur multiplier (H₂) = C₁

Aliases

cyclic	C₄₂₃
abelian	C₉×C₄₇

Subgroups

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₃	C₃×C₄₇
socle, prime radical	C₃×C₄₇	C₃
LCS	G ▷ C₁
UCS	C₁ ◁ G

6 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 56
transitive degree	μ_t(G) = 423

A minimal faithful permutation representation — 1 generator on 56 points:

(1,2,3,4,5,6,7,8,9)(10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56)

Conjugacy classes & character table

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

Relationships

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(423, 1);

center of	C₄₇×C₂₇⋊C₃, C₄₇×#737, C₉×D₅×C₄₇, …
derived subgroup of	D₄₂₃, Dic₄₂₃, D₉×D₄₇, …
Fitting subgroup of	D₉×C₄₇, C₉×D₄₇, D₄₂₃, …
maximal subgroup of	D₉×C₄₇, C₉×D₄₇, D₄₂₃, …
maximal normal subgroup of	D₉×C₄₇, C₉×D₄₇, D₄₂₃, …
maximal quotient of	C₂×C₉×C₄₇, C₂₇×C₄₇, C₃×C₉×C₄₇, …

C9×C47 #88320