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C₂×C₄×C₈ #402

order	64
nilpotency class	1
isoclinism	isoclinic class of C₁
automorphism	Aut(G) = #72219167 (order 2048) · Inn(G) = C₁ · Out(G) = #72219167
homology	Abelianization (H₁) = G · Schur multiplier (H₂) = C₂²×C₄

Aliases

abelian	C₂×C₄×C₈
multiplicative	(ℤ/160)ˣ

Subgroups

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

81 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 14
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 14 points:

(1,2,3,4,5,6,7,8);(9,10,11,12);(13,14)

Conjugacy classes & character table

64 conjugacy classes — view the full character table.

Relationships

center of	#7136, C₈×#97, #7841, …
derived subgroup of	#93776, #93777, #93778, …
Frattini subgroup of	#7102, #7107, #7108, …
maximal subgroup of	#1273, #1277, #1278, …
maximal normal subgroup of	#1273, #1277, #1278, …
maximal quotient of	C₂×C₈², C₂×#322, C₈×M₁₆, …
automorphism group of	C₅×C₃₂
outer automorphism group of	C₅×C₃₂, #63214, C₅×Dic₁₇, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(64, 83);