C₂×D₅₁₅ #14523360

order	2060
isoclinism	isoclinic class of D₅₁₅
automorphism	Aut(G) order 420240
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral	D₁₀₃₀
direct product	C₂×D₅₁₅
unambiguous semidirect product	(C₂×C₅)⋊D₁₀₃
unambiguous semidirect product	(C₂×C₁₀₃)⋊D₅

Splittings

Fitting, socle	C₂×C₅×C₁₀₃ ⋊ C₂
derived, Hall	C₅×C₁₀₃ ⋊ C₂²
Sylow	C₁₀₃ ⋊ C₂×D₅
Sylow	C₅ ⋊ C₂×D₁₀₃

Subgroups

	subgroup	G/H
center, hypercenter	C₂	D₅₁₅
commutator, cosocle, hypocenter	C₅×C₁₀₃	C₂²
Frattini	C₁	G
Fitting, socle	C₂×C₅×C₁₀₃	C₂
LCS	G ▷ C₅×C₁₀₃
UCS	C₁ ◁ C₂

20 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 110
transitive degree	μ_t(G) = 1030

A minimal faithful permutation representation — 2 generators on 110 points:

(1,2)(3,4,5,6,7)(8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110);(4,7)(5,6)(9,110)(10,109)(11,108)(12,107)(13,106)(14,105)(15,104)(16,103)(17,102)(18,101)(19,100)(20,99)(21,98)(22,97)(23,96)(24,95)(25,94)(26,93)(27,92)(28,91)(29,90)(30,89)(31,88)(32,87)(33,86)(34,85)(35,84)(36,83)(37,82)(38,81)(39,80)(40,79)(41,78)(42,77)(43,76)(44,75)(45,74)(46,73)(47,72)(48,71)(49,70)(50,69)(51,68)(52,67)(53,66)(54,65)(55,64)(56,63)(57,62)(58,61)(59,60)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	D₇×D₅₁₅, C₂×D₃₆₀₅, C₂×C₇×D₅₁₅, …
maximal normal subgroup of	C₂×C₇×D₅₁₅, C₂×C₁₁×D₅₁₅, C₂×C₁₃×D₅₁₅, …
maximal quotient of	D₇×D₅₁₅, C₂×D₃₆₀₅, C₂×C₇×D₅₁₅, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(2060, 9);

C2×D515 #14523360