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D₃×C₈² #68045

order	384
isoclinism	isoclinic class of D₃
automorphism	?
homology	Abelianization (H₁) = C₂×C₈² · Schur multiplier (H₂) = C₂²×C₈

Aliases

D₃×C₈²

Splittings

Fitting	C₃×C₈² ⋊ C₂
derived, Sylow	C₃ ⋊ C₂×C₈²

Subgroups

	subgroup	G/H
center, hypercenter	C₈²	D₃
commutator, hypocenter	C₃	C₂×C₈²
Frattini	C₄²	C₂²×D₃
Fitting	C₃×C₈²	C₂
socle	C₂²×C₃	C₂×C₄²
prime radical	C₂²×D₃	C₄²
cosocle	C₃×C₄²	C₂³
LCS	G ▷ C₃
UCS	C₁ ◁ C₈²

280 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 19
transitive degree	μ_t(G) = 192

A minimal faithful permutation representation — 3 generators on 19 points:

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

Conjugacy classes & character table

192 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×C₈×C₁₆, D₃×#1311, C₈×#798, …
maximal normal subgroup of	D₃×C₈×C₁₆, D₃×#1311, C₈×#798, …
maximal quotient of	C₈²×Dic₃, D₃×#1273, #10685609, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 621);