Finite Group Databasebeta

← Prev (#5148) Next (#5150) →

Dic₃×D₈ #5149

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) = C₂³

Aliases

Dic₃×D₈

Splittings

C₃ ⋊ C₄×D₈

Subgroups

	subgroup	G/H
center	C₂²	D₃×D₄
commutator	C₃×C₄	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₃
Fitting, prime radical	C₂×C₃×D₈	C₂
socle	C₂²×C₃	C₂×D₄
cosocle	C₂×C₃×C₄	C₂³
hypercenter	C₂×D₈	D₃
hypocenter	C₃	C₄×D₈
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×D₈

134 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 15
transitive degree	μ_t(G) = 96

A minimal faithful permutation representation — 3 generators on 15 points:

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

Conjugacy classes & character table

42 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#69174, #69175, #69176, …
maximal normal subgroup of	#69174, #69175, #69176, …
maximal quotient of	Dic₃×#102, #70155, #70168, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 708);