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C₃×#216 #3740

order	144
isoclinism	isoclinic class of #213
automorphism	Aut(G) = C₂²×D₃×D₄ (order 192) · Inn(G) = #67 · Out(G) = C₂³
homology	Abelianization (H₁) = C₂²×C₃ · Schur multiplier (H₂) = C₁

Aliases

Splittings

C₃² ⋊ Q₁₆

Subgroups

	subgroup	G/H
center	C₂×C₃	#67
commutator, cosocle	C₃×C₄	C₂²×C₃
Frattini	C₄	C₂×C₃×D₃
Fitting	C₃²×Q₈	C₂
socle, prime radical	C₂×C₃²	D₄
hypercenter	C₃×Q₈	D₃
hypocenter	C₃	C₃×Q₁₆
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₃ ◁ C₃×C₄ ◁ C₃×Q₈

42 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 48

A minimal faithful permutation representation — 2 generators on 22 points:

(1,2,3)(4,5,6)(7,17,11,9)(8,21,14,12)(10,16,20,19)(13,15,18,22);(5,6)(7,14,11,8)(9,22,17,15)(10,13,20,18)(12,16,21,19)

Conjugacy classes & character table

36 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#216, #63943, #63944, …
maximal normal subgroup of	D₃×#216, #63943, #63944, …
maximal quotient of	C₃×#807, C₃×#810, C₃×#835, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(144, 83);