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C₂×C₄⋊C₁₆ #2149

order	128
nilpotency class	2
isoclinism	isoclinic class of D₄
automorphism	Aut(G) = C₂×#72214599 (order 2048) · Inn(G) = C₂² · Out(G) = C₂²×#3025
homology	Abelianization (H₁) = C₂²×C₁₆ · Schur multiplier (H₂) = C₂³

Aliases

C₂×C₄⋊C₁₆

Subgroups

	subgroup	G/H
center	C₂²×C₈	C₂²
commutator	C₂	C₂²×C₁₆
Frattini, cosocle	C₂×C₈	C₂³
socle, prime radical	C₂³	C₂×C₈
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

92 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 22 points:

(1,2);(4,6)(7,8,9,12,10,13,15,18,11,14,16,19,17,20,21,22);(3,4,5,6)

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

Frattini subgroup of	#93687, #93688, #93695, …
maximal subgroup of	#7151, #7153, #7167, …
maximal normal subgroup of	#7151, #7153, #7167, …
maximal quotient of	C₂×#1367, C₂×#1370, #12012, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(128, 881);