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C₃×#204 #3728

order	144
isoclinism	isoclinic class of #204
automorphism	Aut(G) = C₂²×D₃×D₄ (order 192) · Inn(G) = D₁₂ · Out(G) = C₂³
homology	Abelianization (H₁) = C₂²×C₃ · Schur multiplier (H₂) = C₁

Aliases

Splittings

Fitting	C₃²×C₈ ⋊ C₂
Sylow	C₃² ⋊ SD₁₆

Subgroups

	subgroup	G/H
center	C₂×C₃	D₁₂
commutator, cosocle	C₃×C₄	C₂²×C₃
Frattini	C₄	C₂×C₃×D₃
Fitting	C₃²×C₈	C₂
socle	C₂×C₃²	D₄
prime radical	C₃×D₁₂	C₂
hypercenter	C₃×C₈	D₃
hypocenter	C₃	C₃×SD₁₆
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₃ ◁ C₃×C₄ ◁ C₃×C₈

44 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 14
transitive degree	μ_t(G) = 48

A minimal faithful permutation representation — 2 generators on 14 points:

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

Conjugacy classes & character table

45 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#204, #63794, #63796, …
maximal normal subgroup of	D₃×#204, #63794, #63796, …
maximal quotient of	C₃×#816, C₃×#817, C₃×#821, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(144, 71);