D₁₃₁₁₅ #39910879

order	26230
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₁₃₁₁₅
unambiguous semidirect product	C₅⋊D₂₆₂₃
unambiguous semidirect product	C₄₃⋊D₃₀₅
unambiguous semidirect product	C₆₁⋊D₂₁₅

Splittings

Fitting, socle, derived, Hall	C₅×C₄₃×C₆₁ ⋊ C₂
Hall	C₄₃×C₆₁ ⋊ D₅
Hall	C₅×C₆₁ ⋊ D₄₃
Hall	C₅×C₄₃ ⋊ D₆₁
Sylow	C₆₁ ⋊ D₂₁₅
Sylow	C₄₃ ⋊ D₃₀₅
Sylow	C₅ ⋊ D₂₆₂₃

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₅×C₄₃×C₆₁	C₂
LCS	G ▷ C₅×C₄₃×C₆₁
UCS	C₁

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 109
transitive degree	μ_t(G) = 13115

A minimal faithful permutation representation — 2 generators on 109 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)(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109);(2,5)(3,4)(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)(50,109)(51,108)(52,107)(53,106)(54,105)(55,104)(56,103)(57,102)(58,101)(59,100)(60,99)(61,98)(62,97)(63,96)(64,95)(65,94)(66,93)(67,92)(68,91)(69,90)(70,89)(71,88)(72,87)(73,86)(74,85)(75,84)(76,83)(77,82)(78,81)(79,80)

Polycyclic presentation:

2,5,43,61|b^a=b^4,c^a=c^42,d^a=d^60

Conjugacy classes & character table

6559 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂×D₁₃₁₁₅
maximal normal subgroup of	C₂×D₁₃₁₁₅
maximal quotient of	Dic₁₃₁₁₅, C₂×D₁₃₁₁₅

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(26230, 9);

D13115 #39910879