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C₂²×D₃×#120 #11701816

order	768
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁶

Aliases

C₂²×D₃×#120

Splittings

Fitting	C₂²×C₃×#120 ⋊ C₂
Sylow	C₃ ⋊ C₂³×#120

Subgroups

	subgroup	G/H
center	C₂⁴	C₂³×D₃
commutator, cosocle	C₂²×C₃	C₂⁶
Frattini	C₂²	C₂⁵×D₃
Fitting	C₂²×C₃×#120	C₂
socle	C₂⁴×C₃	C₂⁴
hypercenter	C₂²×#120	D₃
hypocenter	C₃	C₂³×#120
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂⁴ ◁ C₂²×#120

42862 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 96

Conjugacy classes & character table

168 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×C₃⋊D₃×#120, C₂²×#10595527, D₃×#23900, …
maximal normal subgroup of	D₃×#23900, D₃×#62817, D₃×#15258, …
maximal quotient of	C₂²×C₃⋊D₃×#120, C₂²×#10595527, D₃×#60388, …
automorphism group of	C₄×Dic₆, C₄×D₁₂, #887, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 1076197);