C₂×D₃₀₃ #11925922

order	1212
isoclinism	isoclinic class of D₃₀₃
automorphism	Aut(G) order 121200
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral	D₆₀₆
direct product	C₂×D₃₀₃
unambiguous semidirect product	(C₂×C₃)⋊D₁₀₁
unambiguous semidirect product	(C₂×C₁₀₁)⋊D₃

Splittings

Fitting, socle	C₂×C₃×C₁₀₁ ⋊ C₂
derived, Hall	C₃×C₁₀₁ ⋊ C₂²
Sylow	C₁₀₁ ⋊ C₂×D₃
Sylow	C₃ ⋊ C₂×D₁₀₁

Subgroups

	subgroup	G/H
center, hypercenter	C₂	D₃₀₃
commutator, cosocle, hypocenter	C₃×C₁₀₁	C₂²
Frattini	C₁	G
Fitting, socle	C₂×C₃×C₁₀₁	C₂
LCS	G ▷ C₃×C₁₀₁
UCS	C₁ ◁ C₂

20 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 106
transitive degree	μ_t(G) = 606

A minimal faithful permutation representation — 2 generators on 106 points:

(1,2)(3,4,5)(6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106);(4,5)(7,106)(8,105)(9,104)(10,103)(11,102)(12,101)(13,100)(14,99)(15,98)(16,97)(17,96)(18,95)(19,94)(20,93)(21,92)(22,91)(23,90)(24,89)(25,88)(26,87)(27,86)(28,85)(29,84)(30,83)(31,82)(32,81)(33,80)(34,79)(35,78)(36,77)(37,76)(38,75)(39,74)(40,73)(41,72)(42,71)(43,70)(44,69)(45,68)(46,67)(47,66)(48,65)(49,64)(50,63)(51,62)(52,61)(53,60)(54,59)(55,58)(56,57)

Conjugacy classes & character table

306 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂×D₉₀₉, D₃×D₃₀₃, C₂×C₃⋊D₃₀₃, …
maximal normal subgroup of	C₂×C₃×D₃₀₃, C₂×C₁₁×D₃₀₃, C₂×C₁₃×D₃₀₃, …
maximal quotient of	C₂×D₉₀₉, D₃×D₃₀₃, C₂×C₃⋊D₃₀₃, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1212, 11);

C2×D303 #11925922