C₄×D₃₂ #13649

order	256
nilpotency class	5
isoclinism	isoclinic class of D₃₂
automorphism	Aut(G) order 8192
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₄×D₃₂

Subgroups

	subgroup	G/H
center	C₂×C₄	D₁₆
commutator	C₁₆	C₂²×C₄
Frattini, cosocle	C₂×C₁₆	C₂³
socle	C₂²	C₂×D₁₆
prime radical	C₂×D₃₂	C₂
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ G
LECS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

107 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 36 points:

(1,2,3,4);(5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36);(6,36)(7,35)(8,34)(9,33)(10,32)(11,31)(12,30)(13,29)(14,28)(15,27)(16,26)(17,25)(18,24)(19,23)(20,22)

Conjugacy classes & character table

76 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#94607, #94609, C₈×D₃₂, …
maximal normal subgroup of	#94607, #94609, C₈×D₃₂, …
maximal quotient of	C₈×D₃₂, #151107, #151121, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 6637);

C4×D32 #13649

Aliases

Subgroups

Presentation

Conjugacy classes & character table

Relationships

GAP

C₄×D₃₂ #13649