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Dic₃×AGL(1,ℤ/8) #83929

order	384
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

Dic₃×AGL(1,ℤ/8)

Splittings

C₃ ⋊ C₄×AGL(1,ℤ/8)

Subgroups

	subgroup	G/H
center	C₂²	C₂×D₃×D₄
commutator	C₃×C₄	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₃
Fitting, prime radical	C₂×C₃×AGL(1,ℤ/8)	C₂
socle	C₂²×C₃	C₂²×D₄
cosocle	C₂×C₃×C₄	C₂⁴
hypercenter	C₂×AGL(1,ℤ/8)	D₃
hypocenter	C₃	C₄×AGL(1,ℤ/8)
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×AGL(1,ℤ/8)

548 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 15 points:

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

Conjugacy classes & character table

66 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	Dic₃×#453, #10801064, Dic₃×#454, …
maximal normal subgroup of	Dic₃×#453, #10801064, Dic₃×#454, …
maximal quotient of	C₃⋊C₈×AGL(1,ℤ/8), #10764772, #10764795, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 16505);