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C₂×C₄×C₄⋊C₄ #2269

order	128
nilpotency class	2
isoclinism	isoclinic class of D₄
automorphism	Aut(G) order 131072
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

C₂×C₄×C₄⋊C₄

Subgroups

	subgroup	G/H
center	C₂³×C₄	C₂²
commutator	C₂	C₂²×C₄²
Frattini, cosocle	C₂³	C₂⁴
socle, prime radical	C₂⁴	C₂³
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³ ▷ C₁

464 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 14
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 14 points:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

Frattini subgroup of	#93283, #93336, #93342, …
maximal subgroup of	C₂×#1279, #7892, #7901, …
maximal normal subgroup of	C₂×#1279, #7892, #7901, …
maximal quotient of	C₂×C₄×#95, C₄²×C₄⋊C₄, C₂×#1434, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(128, 1001);