Dic₁₅₆₅₅ #72295409

order	62620
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₅₆₅₅

Splittings

derived, Hall	C₅×C₃₁×C₁₀₁ ⋊ C₄
Hall	C₃₁×C₁₀₁ ⋊ Dic₅
Hall	C₅×C₁₀₁ ⋊ Dic₃₁
Hall	C₅×C₃₁ ⋊ Dic₁₀₁
Sylow	C₁₀₁ ⋊ Dic₁₅₅
Sylow	C₃₁ ⋊ Dic₅₀₅
Sylow	C₅ ⋊ Dic₃₁₃₁

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₁₅₆₅₅
commutator, hypocenter	C₅×C₃₁×C₁₀₁	C₄
Fitting, socle, prime radical, cosocle	C₂×C₅×C₃₁×C₁₀₁	C₂
LCS	G ▷ C₅×C₃₁×C₁₀₁
UCS	C₁ ◁ C₂

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 141
transitive degree	μ_t(G) = 62620

A minimal faithful permutation representation — 2 generators on 141 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)(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141);(1,3,2,4)(6,9)(7,8)(11,40)(12,39)(13,38)(14,37)(15,36)(16,35)(17,34)(18,33)(19,32)(20,31)(21,30)(22,29)(23,28)(24,27)(25,26)(42,141)(43,140)(44,139)(45,138)(46,137)(47,136)(48,135)(49,134)(50,133)(51,132)(52,131)(53,130)(54,129)(55,128)(56,127)(57,126)(58,125)(59,124)(60,123)(61,122)(62,121)(63,120)(64,119)(65,118)(66,117)(67,116)(68,115)(69,114)(70,113)(71,112)(72,111)(73,110)(74,109)(75,108)(76,107)(77,106)(78,105)(79,104)(80,103)(81,102)(82,101)(83,100)(84,99)(85,98)(86,97)(87,96)(88,95)(89,94)(90,93)(91,92)

Polycyclic presentation:

2^2,5,31,101|a^2=b,c^a=c^4,d^a=d^30,e^a=e^100

Conjugacy classes & character table

15658 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, 31, 101]);;
gap> SetPower(coll, 1, F.2);;
gap> SetConjugate(coll, 3, 1, F.3^4);;
gap> SetConjugate(coll, 4, 1, F.4^30);;
gap> SetConjugate(coll, 5, 1, F.5^100);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic15655 #72295409