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C₄×#107 #1775

order	128
nilpotency class	3
isoclinism	isoclinic class of D₈
automorphism	Aut(G) order 8192
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄

Aliases

Subgroups

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

140 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 16
transitive degree	μ_t(G) = 128

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

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#7171, #7178, #7296, …
maximal normal subgroup of	#7171, #7178, #7296, …
maximal quotient of	C₄×#335, #8274, #8278, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(128, 507);