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C₄×C₂₃ #786

order	92
nilpotency class	1
isoclinism	isoclinic class of C₁
automorphism	Aut(G) = C₂²×C₁₁ (order 44) · Inn(G) = C₁ · Out(G) = C₂²×C₁₁
holomorph	G ⋊ Aut(G) = D₄×AGL(1,𝔽₂₃)
homology	Abelianization (H₁) = G · Schur multiplier (H₂) = C₁

Aliases

cyclic	C₉₂
abelian	C₄×C₂₃

Subgroups

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂	C₂×C₂₃
socle, prime radical	C₂×C₂₃	C₂
LCS	G ▷ C₁
UCS	C₁ ◁ G

6 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 27
transitive degree	μ_t(G) = 92

A minimal faithful permutation representation — 1 generator on 27 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)

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

center of	M₁₆×C₂₃, #41×C₂₃, C₂₃×C₃⋊C₈, …
derived subgroup of	#67263, D₁₈₄, Dic₉₂, …
Frattini subgroup of	Dic₆₃₄₈, Dic₁₀₅₈₀, D₂₁₁₆₀, …
Fitting subgroup of	C₂₃⋊C₈, Dic₄₆, C₄×D₂₃, …
maximal subgroup of	C₂₃⋊C₈, C₈×C₂₃, Dic₄₆, …
maximal normal subgroup of	C₂₃⋊C₈, C₈×C₂₃, Dic₄₆, …
maximal quotient of	C₈×C₂₃, C₂×C₄×C₂₃, Dic₃×C₂₃, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(92, 2);