Dic₁₄₆₄₅ #72291506

order	58580
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₄₆₄₅

Splittings

derived, Hall	C₅×C₂₉×C₁₀₁ ⋊ C₄
Hall	C₂₉×C₁₀₁ ⋊ Dic₅
Hall	C₅×C₁₀₁ ⋊ Dic₂₉
Hall	C₅×C₂₉ ⋊ Dic₁₀₁
Sylow	C₁₀₁ ⋊ Dic₁₄₅
Sylow	C₂₉ ⋊ Dic₅₀₅
Sylow	C₅ ⋊ Dic₂₉₂₉

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₁₄₆₄₅
commutator, hypocenter	C₅×C₂₉×C₁₀₁	C₄
Fitting, socle, prime radical, cosocle	C₂×C₅×C₂₉×C₁₀₁	C₂
LCS	G ▷ C₅×C₂₉×C₁₀₁
UCS	C₁ ◁ C₂

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 139
transitive degree	μ_t(G) = 58580

A minimal faithful permutation representation — 2 generators on 139 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)(39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139);(1,3,2,4)(6,9)(7,8)(11,38)(12,37)(13,36)(14,35)(15,34)(16,33)(17,32)(18,31)(19,30)(20,29)(21,28)(22,27)(23,26)(24,25)(40,139)(41,138)(42,137)(43,136)(44,135)(45,134)(46,133)(47,132)(48,131)(49,130)(50,129)(51,128)(52,127)(53,126)(54,125)(55,124)(56,123)(57,122)(58,121)(59,120)(60,119)(61,118)(62,117)(63,116)(64,115)(65,114)(66,113)(67,112)(68,111)(69,110)(70,109)(71,108)(72,107)(73,106)(74,105)(75,104)(76,103)(77,102)(78,101)(79,100)(80,99)(81,98)(82,97)(83,96)(84,95)(85,94)(86,93)(87,92)(88,91)(89,90)

Polycyclic presentation:

2^2,5,29,101|a^2=b,c^a=c^4,d^a=d^28,e^a=e^100

Conjugacy classes & character table

14648 conjugacy classes. The character table is too large to display.

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e");;
gap> coll := SingleCollector(F, [2, 2, 5, 29, 101]);;
gap> SetPower(coll, 1, F.2);;
gap> SetConjugate(coll, 3, 1, F.3^4);;
gap> SetConjugate(coll, 4, 1, F.4^28);;
gap> SetConjugate(coll, 5, 1, F.5^100);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic14645 #72291506