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C₂×C₂₃ #197

order	46
nilpotency class	1
isoclinism	isoclinic class of C₁
automorphism	Aut(G) = C₂×C₁₁ (order 22) · Inn(G) = C₁ · Out(G) = C₂×C₁₁
holomorph	G ⋊ Aut(G) = C₂×AGL(1,𝔽₂₃)
homology	Abelianization (H₁) = G · Schur multiplier (H₂) = C₁

Aliases

cyclic	C₄₆
abelian	C₂×C₂₃
multiplicative	(ℤ/47)ˣ, (ℤ/94)ˣ

Subgroups

	subgroup	G/H
center	G	C₁
commutator, Frattini	C₁	G
LCS	G ▷ C₁
UCS	C₁ ◁ G

4 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 25
transitive degree	μ_t(G) = 46

A minimal faithful permutation representation — 1 generator on 25 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)

Conjugacy classes & character table

46 conjugacy classes — view the full character table.

Relationships

center of	D₄×C₂₃, Q₈×C₂₃, Dic₃×C₂₃, …
derived subgroup of	Dic₄₆, D₉₂, C₂²⋊D₂₃, …
Frattini subgroup of	C₄×C₅₂₉, Dic₅₂₉, C₄×C₇×C₅₂₉, …
Fitting subgroup of	Dic₂₃, C₂×D₂₃, C₂×C₂₃⋊C₁₁, …
socle of	Dic₂₃, C₄×C₂₃, C₂×D₂₃, …
maximal subgroup of	Dic₂₃, C₄×C₂₃, C₂×D₂₃, …
maximal normal subgroup of	Dic₂₃, C₄×C₂₃, C₂×D₂₃, …
maximal quotient of	C₄×C₂₃, C₂²×C₂₃, D₃×C₂₃, …
automorphism group of	C₄₇, C₂×C₄₇
outer automorphism group of	C₄₇, C₂×C₄₇, Dic₄₇, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(46, 2);