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C₂×C₂²⋊C₈ #406

order	64
nilpotency class	2
isoclinism	isoclinic class of D₄
automorphism	Aut(G) = C₂×#347042 (order 1024) · Inn(G) = C₂² · Out(G) = C₂²×B(4,𝔽₂)
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

C₂×C₂²⋊C₈

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²
commutator	C₂	C₂²×C₈
Frattini, cosocle	C₂×C₄	C₂³
socle	C₂³	C₂×C₄
prime radical	C₂⁴	C₄
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁

101 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 14
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 3 generators on 14 points:

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

Conjugacy classes & character table

40 conjugacy classes — view the full character table.

Relationships

derived subgroup of	#93205, #93206, #93207, …
Frattini subgroup of	#7054, #7055, #7058, …
maximal subgroup of	#1280, #1282, #1289, …
maximal normal subgroup of	#1280, #1282, #1289, …
maximal quotient of	C₂×#323, C₂×#324, #1458, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(64, 87);