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C₂×#4531 #71577

order	384
isoclinism	isoclinic class of #4466
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

Splittings

C₃ ⋊ C₂×H(3,ℤ/4)

Subgroups

	subgroup	G/H
center	C₂×C₄	C₄×Dic₃
commutator	C₃×C₄	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₃
Fitting	C₂×C₃×#117	C₂
socle	C₂²×C₃	#95
prime radical	C₂⁴×C₃	D₄
cosocle	C₂²×C₃×C₄	C₂³
hypercenter	C₂×#117	D₃
hypocenter	C₃	C₂×H(3,ℤ/4)
LCS	G ▷ C₃×C₄ ▷ C₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#117

404 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 21
transitive degree	μ_t(G) = 96

A minimal faithful permutation representation — 3 generators on 21 points:

(3,4,5)(7,11,13,14)(8,15)(12,20,21,19)(16,18);(4,5)(6,7,8,12)(9,14,16,20)(10,13,15,21)(11,18,19,17);(1,2)

Conjugacy classes & character table

84 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	Dic₃×H(3,ℤ/4), C₄×#4531, #10700355, …
maximal normal subgroup of	Dic₃×H(3,ℤ/4), C₄×#4531, #10700355, …
maximal quotient of	C₂×#67737, C₂×#67614, #10684585, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 4153);