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C₂×C₃₂ #369

order	64
nilpotency class	1
isoclinism	isoclinic class of C₁
automorphism	Aut(G) = C₂×#131 (order 64) · Inn(G) = C₁ · Out(G) = C₂×#131
holomorph	G ⋊ Aut(G) = #72225134
homology	Abelianization (H₁) = G · Schur multiplier (H₂) = C₂

Aliases

abelian	C₂×C₃₂
multiplicative	(ℤ/128)ˣ

Subgroups

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₁₆	C₂²
socle, prime radical	C₂²	C₁₆
LCS	G ▷ C₁
UCS	C₁ ◁ G
LECS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

17 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 34
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 2 generators on 34 points:

(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32);(33,34)

Conjugacy classes & character table

64 conjugacy classes — view the full character table.

Relationships

center of	#7510, C₂²⋊C₆₄, C₄⋊C₆₄, …
derived subgroup of	#94690, #94691, #94692, …
Frattini subgroup of	C₄×C₆₄, #7510, #7511, …
maximal subgroup of	C₄×C₃₂, #1397, C₂²⋊C₃₂, …
maximal normal subgroup of	C₄×C₃₂, #1397, C₂²⋊C₃₂, …
maximal quotient of	C₄×C₃₂, C₂²⋊C₃₂, C₄⋊C₃₂, …
automorphism group of	C₁₂₈
outer automorphism group of	C₁₂₈, D₁₂₈, Q₂₅₆, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(64, 50);