C₂×D₅₁₇ #14523382

order	2068
isoclinism	isoclinic class of D₅₁₇
automorphism	Aut(G) order 475640
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral	D₁₀₃₄
direct product	C₂×D₅₁₇
unambiguous semidirect product	(C₂×C₁₁)⋊D₄₇
unambiguous semidirect product	(C₂×C₄₇)⋊D₁₁

Splittings

Fitting, socle	C₂×C₁₁×C₄₇ ⋊ C₂
derived, Hall	C₁₁×C₄₇ ⋊ C₂²
Sylow	C₄₇ ⋊ C₂×D₁₁
Sylow	C₁₁ ⋊ C₂×D₄₇

Subgroups

	subgroup	G/H
center, hypercenter	C₂	D₅₁₇
commutator, cosocle, hypocenter	C₁₁×C₄₇	C₂²
Frattini	C₁	G
Fitting, socle	C₂×C₁₁×C₄₇	C₂
LCS	G ▷ C₁₁×C₄₇
UCS	C₁ ◁ C₂

20 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 60
transitive degree	μ_t(G) = 1034

A minimal faithful permutation representation — 2 generators on 60 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);(4,13)(5,12)(6,11)(7,10)(8,9)(15,60)(16,59)(17,58)(18,57)(19,56)(20,55)(21,54)(22,53)(23,52)(24,51)(25,50)(26,49)(27,48)(28,47)(29,46)(30,45)(31,44)(32,43)(33,42)(34,41)(35,40)(36,39)(37,38)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	D₇×D₅₁₇, C₂×D₃₆₁₉, C₂×C₇×D₅₁₇, …
maximal normal subgroup of	C₂×C₇×D₅₁₇, C₂×C₁₁×D₅₁₇, C₂×C₁₃×D₅₁₇, …
maximal quotient of	D₇×D₅₁₇, C₂×D₃₆₁₉, C₂×C₇×D₅₁₇, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(2068, 7);

C2×D517 #14523382