C₂×D₅₂₇ #14523716

order	2108
isoclinism	isoclinic class of D₅₂₇
automorphism	Aut(G) order 505920
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral	D₁₀₅₄
direct product	C₂×D₅₂₇
unambiguous semidirect product	(C₂×C₁₇)⋊D₃₁
unambiguous semidirect product	(C₂×C₃₁)⋊D₁₇

Splittings

Fitting, socle	C₂×C₁₇×C₃₁ ⋊ C₂
derived, Hall	C₁₇×C₃₁ ⋊ C₂²
Sylow	C₃₁ ⋊ C₂×D₁₇
Sylow	C₁₇ ⋊ C₂×D₃₁

Subgroups

	subgroup	G/H
center, hypercenter	C₂	D₅₂₇
commutator, cosocle, hypocenter	C₁₇×C₃₁	C₂²
Frattini	C₁	G
Fitting, socle	C₂×C₁₇×C₃₁	C₂
LCS	G ▷ C₁₇×C₃₁
UCS	C₁ ◁ C₂

20 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 50
transitive degree	μ_t(G) = 1054

A minimal faithful permutation representation — 2 generators on 50 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);(4,19)(5,18)(6,17)(7,16)(8,15)(9,14)(10,13)(11,12)(21,50)(22,49)(23,48)(24,47)(25,46)(26,45)(27,44)(28,43)(29,42)(30,41)(31,40)(32,39)(33,38)(34,37)(35,36)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	D₇×D₅₂₇, C₂×D₃₆₈₉, C₂×C₇×D₅₂₇, …
maximal normal subgroup of	C₂×C₇×D₅₂₇, C₂×C₁₁×D₅₂₇, C₂×C₁₃×D₅₂₇, …
maximal quotient of	D₇×D₅₂₇, C₂×D₃₆₈₉, C₂×C₇×D₅₂₇, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(2108, 9);

C2×D527 #14523716