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C₄×C₁₆ #345

order	64
nilpotency class	1
isoclinism	isoclinic class of C₁
automorphism	Aut(G) = C₂×#2010 (order 256) · Inn(G) = C₁ · Out(G) = C₂×#2010
homology	Abelianization (H₁) = G · Schur multiplier (H₂) = C₄

Aliases

abelian	C₄×C₁₆
multiplicative	(ℤ/85)ˣ, (ℤ/170)ˣ

Subgroups

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂×C₈	C₂²
socle, prime radical	C₂²	C₂×C₈
LCS	G ▷ C₁
UCS	C₁ ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

29 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 2 generators on 20 points:

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

Conjugacy classes & character table

64 conjugacy classes — view the full character table.

Relationships

center of	#7329, #7331, #11399, …
derived subgroup of	#93862, #93863, #93864, …
Frattini subgroup of	C₈×C₃₂, #7329, #7330, …
maximal subgroup of	C₈×C₁₆, #1311, #1312, …
maximal normal subgroup of	C₈×C₁₆, #1311, #1312, …
maximal quotient of	C₈×C₁₆, #1312, #1374, …
automorphism group of	C₅×C₁₇, C₂×C₅×C₁₇
outer automorphism group of	C₅×C₁₇, C₂×C₅×C₁₇, C₅×D₃×C₁₇

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(64, 26);