C₄×SD₆₄ #13650

order	256
nilpotency class	5
isoclinism	isoclinic class of D₃₂
automorphism	Aut(G) = C₂⁴×#12067 (order 4096) · Inn(G) = D₁₆ · Out(G) = C₂⁴×C₈
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₄×SD₆₄

Subgroups

	subgroup	G/H
center	C₂×C₄	D₁₆
commutator	C₁₆	C₂²×C₄
Frattini, cosocle	C₂×C₁₆	C₂³
socle	C₂²	C₂×D₁₆
prime radical	C₂×D₁₆	C₂²
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

101 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);(6,20)(7,35)(8,18)(9,33)(10,16)(11,31)(12,14)(13,29)(15,27)(17,25)(19,23)(22,36)(24,34)(26,32)(28,30)

Conjugacy classes & character table

76 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₈×SD₆₄, #151078, #151108, …
maximal normal subgroup of	C₈×SD₆₄, #151078, #151108, …
maximal quotient of	C₈×SD₆₄, #151108, #151109, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 6638);

C4×SD64 #13650

Aliases

Subgroups

Presentation

Conjugacy classes & character table

Relationships

GAP

C₄×SD₆₄ #13650