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C₂²×C₄²×D₃ #85672

order	384
isoclinism	isoclinic class of D₃
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄² · Schur multiplier (H₂) = C₂⁹×C₄

Aliases

C₂²×C₄²×D₃

Splittings

Fitting	C₂²×C₃×C₄² ⋊ C₂
derived, Sylow	C₃ ⋊ C₂³×C₄²

Subgroups

	subgroup	G/H
center, hypercenter	C₂²×C₄²	D₃
commutator, hypocenter	C₃	C₂³×C₄²
Frattini	C₂²	C₂⁴×D₃
Fitting	C₂²×C₃×C₄²	C₂
socle	C₂⁴×C₃	C₂³
prime radical	C₂⁴×D₃	C₂²
cosocle	C₂²×C₃	C₂⁵
LCS	G ▷ C₃
UCS	C₁ ◁ C₂²×C₄²

3000 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 15
transitive degree	μ_t(G) = 192

A minimal faithful permutation representation — 5 generators on 15 points:

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

Conjugacy classes & character table

192 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10700403, #10700427, C₄×D₃×#95, …
maximal normal subgroup of	#10700403, #10700427, C₄×D₃×#95, …
maximal quotient of	C₂×C₄³×D₃, C₂²×C₄²×Dic₃, C₂²×D₃×#95, …
automorphism group of	C₅×D₃×C₁₆, C₅×#798

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 18248);