C₂×#1906 #20999

order	256
nilpotency class	3
isoclinism	isoclinic class of #471
automorphism	Aut(G) order 8192
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

direct product

C₂×#1906

Subgroups

	subgroup	G/H
center	C₂×C₄	#121
commutator	C₂×C₄	C₂³×C₄
Frattini, cosocle	C₂²×C₄	C₂⁴
socle	C₂²	#386
prime radical	C₂×#575	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

899 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 18
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 4 generators on 18 points:

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

Conjugacy classes & character table

64 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₄×#1906, #354237, #354318, …
maximal normal subgroup of	C₄×#1906, #354237, #354318, …
maximal quotient of	C₂×#9095, C₂×#9096, #196172, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 13987);

C2×#1906 #20999

Aliases

Subgroups

Presentation

Conjugacy classes & character table

Relationships

GAP

C₂×#1906 #20999