C₂×D₅₁₁ #14523291

order	2044
isoclinism	isoclinic class of D₅₁₁
automorphism	Aut(G) order 441504
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral	D₁₀₂₂
direct product	C₂×D₅₁₁
unambiguous semidirect product	(C₂×C₇)⋊D₇₃
unambiguous semidirect product	(C₂×C₇₃)⋊D₇

Splittings

Fitting, socle	C₂×C₇×C₇₃ ⋊ C₂
derived, Hall	C₇×C₇₃ ⋊ C₂²
Sylow	C₇₃ ⋊ C₂×D₇
Sylow	C₇ ⋊ C₂×D₇₃

Subgroups

	subgroup	G/H
center, hypercenter	C₂	D₅₁₁
commutator, cosocle, hypocenter	C₇×C₇₃	C₂²
Frattini	C₁	G
Fitting, socle	C₂×C₇×C₇₃	C₂
LCS	G ▷ C₇×C₇₃
UCS	C₁ ◁ C₂

20 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 82
transitive degree	μ_t(G) = 1022

A minimal faithful permutation representation — 2 generators on 82 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);(4,9)(5,8)(6,7)(11,82)(12,81)(13,80)(14,79)(15,78)(16,77)(17,76)(18,75)(19,74)(20,73)(21,72)(22,71)(23,70)(24,69)(25,68)(26,67)(27,66)(28,65)(29,64)(30,63)(31,62)(32,61)(33,60)(34,59)(35,58)(36,57)(37,56)(38,55)(39,54)(40,53)(41,52)(42,51)(43,50)(44,49)(45,48)(46,47)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂×D₃₅₇₇, D₇×D₅₁₁, C₂×C₇⋊D₅₁₁, …
maximal normal subgroup of	C₂×C₇×D₅₁₁, C₂×C₁₁×D₅₁₁, C₂×C₁₃×D₅₁₁, …
maximal quotient of	C₂×D₃₅₇₇, D₇×D₅₁₁, C₂×C₇⋊D₅₁₁, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(2044, 9);

C2×D511 #14523291