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C₂×C₄×D₈ #2936

order	128
nilpotency class	3
isoclinism	isoclinic class of D₈
automorphism	Aut(G) order 16384
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

C₂×C₄×D₈

Subgroups

	subgroup	G/H
center	C₂²×C₄	D₄
commutator	C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄	C₂⁴
socle	C₂³	C₂×D₄
prime radical	C₂²×D₈	C₂
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁

308 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);(8,14)(9,13)(10,12)

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#8947, #8951, #9657, …
maximal normal subgroup of	#8947, #8951, #9657, …
maximal quotient of	C₂×C₈×D₈, C₂×#1581, C₂×#1589, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(128, 1668);