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C₂×D₃×C₁₆ #4899

order	192
isoclinism	isoclinic class of D₃
automorphism	Aut(G) = C₂×D₃×#525 (order 768) · Inn(G) = D₃ · Out(G) = C₂×#525
homology	Abelianization (H₁) = C₂²×C₁₆ · Schur multiplier (H₂) = C₂³

Aliases

C₂×D₃×C₁₆

Splittings

Fitting	C₂×C₃×C₁₆ ⋊ C₂
derived, Sylow	C₃ ⋊ C₂²×C₁₆

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₁₆	D₃
commutator, hypocenter	C₃	C₂²×C₁₆
Frattini	C₈	C₂²×D₃
Fitting	C₂×C₃×C₁₆	C₂
socle	C₂²×C₃	C₂×C₈
prime radical	C₂²×D₃	C₈
cosocle	C₃×C₈	C₂³
LCS	G ▷ C₃
UCS	C₁ ◁ C₂×C₁₆

98 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 21
transitive degree	μ_t(G) = 96

A minimal faithful permutation representation — 3 generators on 21 points:

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

Conjugacy classes & character table

96 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	(C₂×D₃)⋊C₃₂, C₄×D₃×C₁₆, #69051, …
maximal normal subgroup of	(C₂×D₃)⋊C₃₂, C₄×D₃×C₁₆, #69051, …
maximal quotient of	C₁₆×Dic₆, C₄×D₃×C₁₆, #69052, …
automorphism group of	C₃⋊C₆₄, Dic₃×C₁₇, C₂×D₃×C₁₇, …
outer automorphism group of	C₂²×C₃×C₁₇, #13737

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 458);