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C₂×C₄×M₁₆ #2871

order	128
nilpotency class	2
isoclinism	isoclinic class of D₄
automorphism	Aut(G) order 32768
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

C₂×C₄×M₁₆

Subgroups

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

328 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 14
transitive degree	μ_t(G) = 64

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

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

derived subgroup of	#72338365, #72338368, #72338370, …
Frattini subgroup of	#93284, #93285, #93292, …
maximal subgroup of	#7848, #7850, #7852, …
maximal normal subgroup of	#7848, #7850, #7852, …
maximal quotient of	C₂×C₈×M₁₆, C₂×#1451, C₂×#1453, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(128, 1603);