Finite Group Databasebeta

← Prev (#1575) Next (#1577) →

C₈×SD₁₆ #1576

order	128
nilpotency class	3
isoclinism	isoclinic class of D₈
automorphism	Aut(G) = C₂⁶×D₄ (order 512) · Inn(G) = D₄ · Out(G) = C₂⁶
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂²

Aliases

C₈×SD₁₆

Subgroups

	subgroup	G/H
center	C₂×C₈	D₄
commutator	C₄	C₂²×C₈
Frattini, cosocle	C₄²	C₂³
socle	C₂²	C₄×D₄
prime radical	C₂×D₄	C₂×C₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₈ ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

90 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 16
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 16 points:

(1,2,3,4,5,6,7,8);(9,10,11,12,13,14,15,16);(10,12)(11,15)(14,16)

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₁₆×SD₁₆, #12049, #12050, …
maximal normal subgroup of	C₁₆×SD₁₆, #12049, #12050, …
maximal quotient of	C₈×#102, C₈×#103, C₈×#106, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(128, 308);