D₆₆₉₅ #27211485

order	13390
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₆₆₉₅
unambiguous semidirect product	C₅⋊D₁₃₃₉
unambiguous semidirect product	C₁₃⋊D₅₁₅
unambiguous semidirect product	C₁₀₃⋊D₆₅

Splittings

Fitting, socle, derived, Hall	C₅×C₁₃×C₁₀₃ ⋊ C₂
Hall	C₁₃×C₁₀₃ ⋊ D₅
Hall	C₅×C₁₀₃ ⋊ D₁₃
Sylow	C₁₀₃ ⋊ D₆₅
Hall	C₅×C₁₃ ⋊ D₁₀₃
Sylow	C₁₃ ⋊ D₅₁₅
Sylow	C₅ ⋊ D₁₃₃₉

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₅×C₁₃×C₁₀₃	C₂
LCS	G ▷ C₅×C₁₃×C₁₀₃
UCS	C₁

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 121
transitive degree	μ_t(G) = 6695

A minimal faithful permutation representation — 2 generators on 121 points:

(1,2,3,4,5)(6,7,8,9,10,11,12,13,14,15,16,17,18)(19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121);(2,5)(3,4)(7,18)(8,17)(9,16)(10,15)(11,14)(12,13)(20,121)(21,120)(22,119)(23,118)(24,117)(25,116)(26,115)(27,114)(28,113)(29,112)(30,111)(31,110)(32,109)(33,108)(34,107)(35,106)(36,105)(37,104)(38,103)(39,102)(40,101)(41,100)(42,99)(43,98)(44,97)(45,96)(46,95)(47,94)(48,93)(49,92)(50,91)(51,90)(52,89)(53,88)(54,87)(55,86)(56,85)(57,84)(58,83)(59,82)(60,81)(61,80)(62,79)(63,78)(64,77)(65,76)(66,75)(67,74)(68,73)(69,72)(70,71)

Polycyclic presentation:

2,5,13,103|b^a=b^4,c^a=c^12,d^a=d^102

Conjugacy classes & character table

3349 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₆₅×D₁₀₃, D₁₃×D₅₁₅, D₅×D₁₃₃₉, …
maximal normal subgroup of	D₆₅×D₁₀₃, D₁₃×D₅₁₅, D₅×D₁₃₃₉, …
maximal quotient of	Dic₆₆₉₅, C₂×D₆₆₉₅

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13390, 8);

D6695 #27211485