Dic₉₁₅₉ #49143802

order	36636
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₉₁₅₉
unambiguous semidirect product	C₃⋊Dic₃₀₅₃
unambiguous semidirect product	C₄₃⋊Dic₂₁₃
unambiguous semidirect product	C₇₁⋊Dic₁₂₉

Splittings

derived, Hall	C₃×C₄₃×C₇₁ ⋊ C₄
Hall	C₄₃×C₇₁ ⋊ Dic₃
Hall	C₃×C₇₁ ⋊ Dic₄₃
Hall	C₃×C₄₃ ⋊ Dic₇₁
Sylow	C₇₁ ⋊ Dic₁₂₉
Sylow	C₄₃ ⋊ Dic₂₁₃
Sylow	C₃ ⋊ Dic₃₀₅₃

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₉₁₅₉
commutator, hypocenter	C₃×C₄₃×C₇₁	C₄
Fitting, socle, prime radical, cosocle	C₂×C₃×C₄₃×C₇₁	C₂
LCS	G ▷ C₃×C₄₃×C₇₁
UCS	C₁ ◁ C₂

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 121
transitive degree	μ_t(G) = 36636

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);(1,3,2,4)(6,7)(9,50)(10,49)(11,48)(12,47)(13,46)(14,45)(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)(52,121)(53,120)(54,119)(55,118)(56,117)(57,116)(58,115)(59,114)(60,113)(61,112)(62,111)(63,110)(64,109)(65,108)(66,107)(67,106)(68,105)(69,104)(70,103)(71,102)(72,101)(73,100)(74,99)(75,98)(76,97)(77,96)(78,95)(79,94)(80,93)(81,92)(82,91)(83,90)(84,89)(85,88)(86,87)

Polycyclic presentation:

2,3,43,71,2|a^2=e,b^a=b^2,c^a=c^42,d^a=d^70

Conjugacy classes & character table

9162 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(36636, 9);

Dic9159 #49143802