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C₂×D₃×B(4,𝔽₂) #11661898

order	768
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹¹

Aliases

C₂×D₃×B(4,𝔽₂)

Splittings

Fitting	C₂×C₃×B(4,𝔽₂) ⋊ C₂
Sylow	C₃ ⋊ C₂²×B(4,𝔽₂)

Subgroups

	subgroup	G/H
center	C₂²	D₃×#120
commutator, cosocle	C₂³×C₃	C₂⁵
Frattini	C₂³	C₂⁴×D₃
Fitting	C₂×C₃×B(4,𝔽₂)	C₂
socle	C₂²×C₃	C₂×#120
hypercenter	C₂×B(4,𝔽₂)	D₃
hypocenter	C₃	C₂²×B(4,𝔽₂)
LCS	G ▷ C₂³×C₃ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂×B(4,𝔽₂)

7014 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 48

Conjugacy classes & character table

96 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×C₃⋊D₃×B(4,𝔽₂), D₃×#23900, C₂×D₃×#1020⋊C₂, …
maximal normal subgroup of	D₃×#23900, D₃×#33551, D₃×#13343, …
maximal quotient of	C₂×C₃⋊D₃×B(4,𝔽₂), D₃×#23900, D₃×#33551, …
automorphism group of	C₂×#208, C₂×C₃×Dic₆, C₂×C₃×D₁₂, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 1036279);