C₂×D₃₆₉ #13093275

order	1476
isoclinism	isoclinic class of D₃₆₉
automorphism	Aut(G) order 177120
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral	D₇₃₈
direct product	C₂×D₃₆₉
unambiguous semidirect product	(C₂×C₉)⋊D₄₁
unambiguous semidirect product	(C₂×C₄₁)⋊D₉

Splittings

Fitting	C₂×C₉×C₄₁ ⋊ C₂
derived, Hall	C₉×C₄₁ ⋊ C₂²
Sylow	C₄₁ ⋊ C₂×D₉
Sylow	C₉ ⋊ C₂×D₄₁

Subgroups

	subgroup	G/H
center, hypercenter	C₂	D₃₆₉
commutator, cosocle, hypocenter	C₉×C₄₁	C₂²
Frattini	C₃	C₂×D₁₂₃
Fitting	C₂×C₉×C₄₁	C₂
socle	C₂×C₃×C₄₁	D₃
LCS	G ▷ C₉×C₄₁
UCS	C₁ ◁ C₂

30 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 52
transitive degree	μ_t(G) = 738

A minimal faithful permutation representation — 2 generators on 52 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);(4,11)(5,10)(6,9)(7,8)(13,52)(14,51)(15,50)(16,49)(17,48)(18,47)(19,46)(20,45)(21,44)(22,43)(23,42)(24,41)(25,40)(26,39)(27,38)(28,37)(29,36)(30,35)(31,34)(32,33)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	D₁₁×D₃₆₉, C₂×D₄₀₅₉, C₂×C₁₁×D₃₆₉, …
maximal normal subgroup of	C₂×C₁₁×D₃₆₉, C₂×C₁₃×D₃₆₉, C₂×C₁₇×D₃₆₉, …
maximal quotient of	D₁₁×D₃₆₉, C₂×D₄₀₅₉, C₂×C₁₁×D₃₆₉, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1476, 11);

C2×D369 #13093275