Finite Group Databasebeta

← Prev (#1759) Next (#1761) →

C₄×#102 #1760

order	128
nilpotency class	3
isoclinism	isoclinic class of D₈
automorphism	Aut(G) order 4096
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

Subgroups

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

198 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)(11,13)(12,15)(14,16);(10,13)(11,16)(14,15)

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₈×#102, #8200, #8202, …
maximal normal subgroup of	C₈×#102, #8200, #8202, …
maximal quotient of	C₄×D₄⋊C₈, #8174, #8177, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(128, 492);