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C₂×C₄×H(3,ℤ/4) #353122

order	512
nilpotency class	2
isoclinism	isoclinic class of H(3,ℤ/4)
automorphism	?
homology	Abelianization (H₁) = C₂×C₄³ · Schur multiplier (H₂) = C₂⁴×C₄³

Aliases

C₂×C₄×H(3,ℤ/4)

Subgroups

	subgroup	G/H
center	C₂×C₄²	C₄²
commutator	C₄	C₂×C₄³
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂³	C₂×#95
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₄ ▷ C₁
UCS	C₁ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2790 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 22 points:

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

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×C₄×#2010, #72213046, C₂×#128834, …
maximal normal subgroup of	C₂×C₄×#2010, #72213046, C₂×#128834, …
maximal quotient of	C₂×#105252, C₂×#105255, C₂×#123426

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 260318);