Finite Group Databasebeta

← Prev (#5180) Next (#5182) →

Dic₃×Q₁₆ #5181

order	192
automorphism	Aut(G) = C₂²×D₃×#3003 (order 3072) · Inn(G) = D₃×D₄ · Out(G) = C₂³×D₄
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

Dic₃×Q₁₆

Splittings

C₃ ⋊ C₄×Q₁₆

Subgroups

	subgroup	G/H
center	C₂²	D₃×D₄
commutator	C₃×C₄	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₃
Fitting	C₂×C₃×Q₁₆	C₂
socle, prime radical	C₂²×C₃	C₂×D₄
cosocle	C₂×C₃×C₄	C₂³
hypercenter	C₂×Q₁₆	D₃
hypocenter	C₃	C₄×Q₁₆
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×Q₁₆

110 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 23
transitive degree	μ_t(G) = 192

A minimal faithful permutation representation — 3 generators on 23 points:

(5,7,9,15)(6,10,12,19)(8,17,18,14)(11,20,16,13)(21,22,23);(5,6,9,12)(7,13,15,20)(8,16,18,11)(10,17,19,14);(1,2,3,4)(22,23)

Conjugacy classes & character table

42 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#69206, #69207, #69208, …
maximal normal subgroup of	#69206, #69207, #69208, …
maximal quotient of	Dic₃×#103, #70224, #70236, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 740);