C₂×D₃₄₁ #13062891

order	1364
isoclinism	isoclinic class of D₃₄₁
automorphism	Aut(G) order 204600
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral	D₆₈₂
direct product	C₂×D₃₄₁
unambiguous semidirect product	(C₂×C₁₁)⋊D₃₁
unambiguous semidirect product	(C₂×C₃₁)⋊D₁₁

Splittings

Fitting, socle	C₂×C₁₁×C₃₁ ⋊ C₂
derived, Hall	C₁₁×C₃₁ ⋊ C₂²
Sylow	C₃₁ ⋊ C₂×D₁₁
Sylow	C₁₁ ⋊ C₂×D₃₁

Subgroups

	subgroup	G/H
center, hypercenter	C₂	D₃₄₁
commutator, cosocle, hypocenter	C₁₁×C₃₁	C₂²
Frattini	C₁	G
Fitting, socle	C₂×C₁₁×C₃₁	C₂
LCS	G ▷ C₁₁×C₃₁
UCS	C₁ ◁ C₂

20 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 682

A minimal faithful permutation representation — 2 generators on 44 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);(4,13)(5,12)(6,11)(7,10)(8,9)(15,44)(16,43)(17,42)(18,41)(19,40)(20,39)(21,38)(22,37)(23,36)(24,35)(25,34)(26,33)(27,32)(28,31)(29,30)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂×C₁₁⋊(D₃₁⋊C₃), D₃×D₃₄₁, C₂×C₃×D₃₄₁, …
maximal normal subgroup of	C₂×C₁₁⋊(D₃₁⋊C₃), C₂×C₃×D₃₄₁, C₂×C₁₁×D₃₄₁, …
maximal quotient of	D₃×D₃₄₁, C₂×C₃×D₃₄₁, C₂×D₁₀₂₃, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1364, 8);

C2×D341 #13062891