Finite Group Databasebeta

← Prev (#796) Next (#798) →

D₃×C₁₆ #797

order	96
isoclinism	isoclinic class of D₃
automorphism	Aut(G) = C₂²×C₄×D₃ (order 96) · Inn(G) = D₃ · Out(G) = C₂²×C₄
homology	Abelianization (H₁) = C₂×C₁₆ · Schur multiplier (H₂) = C₂

Aliases

D₃×C₁₆

Splittings

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

Subgroups

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

28 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 19
transitive degree	μ_t(G) = 48

A minimal faithful permutation representation — 2 generators on 19 points:

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

Conjugacy classes & character table

48 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×C₃₂, #4447, C₂×D₃×C₁₆, …
maximal normal subgroup of	D₃×C₃₂, #4447, C₂×D₃×C₁₆, …
maximal quotient of	D₃×C₃₂, #4447, Dic₃×C₁₆, …
automorphism group of	C₂²×C₁₇, D₃×C₁₇
outer automorphism group of	C₂²×C₁₇, Q₈×C₁₇, C₁₇×3₋¹⁺²

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(96, 4);