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D₃×D₄² #86619

order	384
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹⁰

Aliases

D₃×D₄²

Splittings

Fitting	C₃×D₄² ⋊ C₂
Sylow	C₃ ⋊ C₂×D₄²

Subgroups

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₃
commutator, socle, cosocle	C₂²×C₃	C₂⁵
Fitting	C₃×D₄²	C₂
hypercenter	D₄²	D₃
hypocenter	C₃	C₂×D₄²
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ D₄²

2536 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 11
transitive degree	μ_t(G) = 48

A minimal faithful permutation representation — 5 generators on 11 points:

(1,2,3)(9,11);(8,9)(10,11);(5,7);(4,5)(6,7);(2,3)

Conjugacy classes & character table

75 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#1619, D₃×#1629, D₃×#1639, …
maximal normal subgroup of	D₃×#1619, D₃×#1629, D₃×#1639, …
maximal quotient of	D₃×D₄×C₂²⋊C₄, D₃×D₄×C₄⋊C₄, D₃×#2358, …
automorphism group of	#823, #824, #908, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 19195);