C₂×D₅₃₉ #14524154

order	2156
isoclinism	isoclinic class of D₅₃₉
automorphism	Aut(G) order 452760
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral	D₁₀₇₈
direct product	C₂×D₅₃₉
unambiguous semidirect product	(C₂×C₁₁)⋊D₄₉
unambiguous semidirect product	(C₂×C₄₉)⋊D₁₁

Splittings

Fitting	C₂×C₁₁×C₄₉ ⋊ C₂
derived, Hall	C₁₁×C₄₉ ⋊ C₂²
Sylow	C₄₉ ⋊ C₂×D₁₁
Sylow	C₁₁ ⋊ C₂×D₄₉

Subgroups

	subgroup	G/H
center, hypercenter	C₂	D₅₃₉
commutator, cosocle, hypocenter	C₁₁×C₄₉	C₂²
Frattini	C₇	C₂×D₇₇
Fitting	C₂×C₁₁×C₄₉	C₂
socle	C₂×C₇×C₁₁	D₇
LCS	G ▷ C₁₁×C₄₉
UCS	C₁ ◁ C₂

30 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 62
transitive degree	μ_t(G) = 1078

A minimal faithful permutation representation — 2 generators on 62 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);(4,51)(5,50)(6,49)(7,48)(8,47)(9,46)(10,45)(11,44)(12,43)(13,42)(14,41)(15,40)(16,39)(17,38)(18,37)(19,36)(20,35)(21,34)(22,33)(23,32)(24,31)(25,30)(26,29)(27,28)(53,62)(54,61)(55,60)(56,59)(57,58)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂×D₅₉₂₉, D₁₁×D₅₃₉, C₂×C₁₁⋊D₅₃₉, …
maximal normal subgroup of	C₂×C₁₁×D₅₃₉, C₂×C₁₃×D₅₃₉, C₂×C₁₇×D₅₃₉, …
maximal quotient of	C₂×D₅₉₂₉, D₁₁×D₅₃₉, C₂×C₁₁⋊D₅₃₉, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(2156, 9);

C2×D539 #14524154