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C₄×#117 #2270

order	128
nilpotency class	2
isoclinism	isoclinic class of D₄
automorphism	Aut(G) order 65536
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂³×C₄²

Aliases

Subgroups

	subgroup	G/H
center	C₂×C₄²	C₂²
commutator	C₂	C₂²×C₄²
Frattini, socle, cosocle	C₂³	C₂⁴
prime radical	C₂⁴	C₂³
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂³ ▷ C₁

376 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 16 points:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

derived subgroup of	#72257880, #72265296, #72265555, …
Frattini subgroup of	#93290, #93291, #93294, …
maximal subgroup of	#7854, #7856, #7866, …
maximal normal subgroup of	#7854, #7856, #7866, …
maximal quotient of	C₄×#376, C₄²×C₂²⋊C₄, C₄²×C₄⋊C₄, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(128, 1002);