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C₂×C₄×C₁₆ #2105

order	128
nilpotency class	1
isoclinism	isoclinic class of C₁
automorphism	Aut(G) = C₂×#72238006 (order 4096) · Inn(G) = C₁ · Out(G) = C₂×#72238006
homology	Abelianization (H₁) = G · Schur multiplier (H₂) = C₂²×C₄

Aliases

abelian	C₂×C₄×C₁₆
multiplicative	(ℤ/255)ˣ, (ℤ/272)ˣ, (ℤ/320)ˣ, (ℤ/340)ˣ, (ℤ/510)ˣ

Subgroups

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂×C₈	C₂³
socle, prime radical	C₂³	C₂×C₈
LCS	G ▷ C₁
UCS	C₁ ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

108 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 22 points:

(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16);(17,18,19,20);(21,22)

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

center of	#93989, #126519, C₁₆×#97, …
Frattini subgroup of	#93886, #93889, #93890, …
maximal subgroup of	#7136, #7141, #7154, …
maximal normal subgroup of	#7136, #7141, #7154, …
maximal quotient of	C₂×C₈×C₁₆, C₂×#1312, C₁₆×M₁₆, …
automorphism group of	C₃×C₅×C₁₇, C₁₆×C₁₇, C₅×C₆₄, …
outer automorphism group of	C₃×C₅×C₁₇, C₁₆×C₁₇, C₅×C₆₄, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(128, 837);