C₂×D₅₃₁ #14523770

order	2124
isoclinism	isoclinic class of D₅₃₁
automorphism	Aut(G) order 369576
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral	D₁₀₆₂
direct product	C₂×D₅₃₁
unambiguous semidirect product	(C₂×C₉)⋊D₅₉
unambiguous semidirect product	(C₂×C₅₉)⋊D₉

Splittings

Fitting	C₂×C₉×C₅₉ ⋊ C₂
derived, Hall	C₉×C₅₉ ⋊ C₂²
Sylow	C₅₉ ⋊ C₂×D₉
Sylow	C₉ ⋊ C₂×D₅₉

Subgroups

	subgroup	G/H
center, hypercenter	C₂	D₅₃₁
commutator, cosocle, hypocenter	C₉×C₅₉	C₂²
Frattini	C₃	C₂×D₁₇₇
Fitting	C₂×C₉×C₅₉	C₂
socle	C₂×C₃×C₅₉	D₃
LCS	G ▷ C₉×C₅₉
UCS	C₁ ◁ C₂

30 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 70
transitive degree	μ_t(G) = 1062

A minimal faithful permutation representation — 2 generators on 70 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);(4,11)(5,10)(6,9)(7,8)(13,70)(14,69)(15,68)(16,67)(17,66)(18,65)(19,64)(20,63)(21,62)(22,61)(23,60)(24,59)(25,58)(26,57)(27,56)(28,55)(29,54)(30,53)(31,52)(32,51)(33,50)(34,49)(35,48)(36,47)(37,46)(38,45)(39,44)(40,43)(41,42)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	D₇×D₅₃₁, C₂×C₇×D₅₃₁, C₂×D₃₇₁₇, …
maximal normal subgroup of	C₂×C₇×D₅₃₁, C₂×C₁₁×D₅₃₁, C₂×C₁₃×D₅₃₁, …
maximal quotient of	D₇×D₅₃₁, C₂×C₇×D₅₃₁, C₂×D₃₇₁₇, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(2124, 10);

C2×D531 #14523770