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C₂×C₄² #114

order	32
nilpotency class	1
isoclinism	isoclinic class of C₁
automorphism	Aut(G) = #72209884 (order 1536) · Inn(G) = C₁ · Out(G) = #72209884
homology	Abelianization (H₁) = G · Schur multiplier (H₂) = C₂²×C₄

Aliases

abelian	C₂×C₄²
multiplicative	(ℤ/80)ˣ

Subgroups

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂²	C₂³
socle, prime radical	C₂³	C₂²
LCS	G ▷ C₁
UCS	C₁ ◁ G
LECS	G ▷ C₂² ▷ C₁

54 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 10
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 3 generators on 10 points:

(1,2,3,4);(5,6,7,8);(9,10)

Conjugacy classes & character table

32 conjugacy classes — view the full character table.

Relationships

center of	#1273, C₂×#322, C₄×#97, …
derived subgroup of	#7402, #7403, #7404, …
Frattini subgroup of	#1270, #1271, #1272, …
Fitting subgroup of	C₂×C₄²⋊C₃, #4623, C₂×#857, …
socle of	#72304317, #41×#525, C₂×#10585995, …
maximal subgroup of	#336, #339, C₄³, …
maximal normal subgroup of	#336, #339, C₄³, …
maximal quotient of	C₄³, C₂×#95, #376, …
automorphism group of	C₅×C₁₆
outer automorphism group of	C₅×C₁₆, D₈₀, Dic₄₀, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(32, 21);