D₂₁₀₀₄ #72274802

order	42008
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₂₁₀₀₄

Splittings

Fitting	C₄×C₅₉×C₈₉ ⋊ C₂
Hall	C₅₉×C₈₉ ⋊ D₄
Sylow	C₈₉ ⋊ D₂₃₆
Sylow	C₅₉ ⋊ D₃₅₆

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₅₂₅₁
commutator, socle, cosocle	C₂×C₅₉×C₈₉	C₂²
Fitting	C₄×C₅₉×C₈₉	C₂
hypercenter	C₄	D₅₂₅₁
hypocenter	C₅₉×C₈₉	D₄
LCS	G ▷ C₂×C₅₉×C₈₉ ▷ C₅₉×C₈₉
UCS	C₁ ◁ C₂ ◁ C₄

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 152
transitive degree	μ_t(G) = 21004

A minimal faithful permutation representation — 2 generators on 152 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)(64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152);(2,4)(6,63)(7,62)(8,61)(9,60)(10,59)(11,58)(12,57)(13,56)(14,55)(15,54)(16,53)(17,52)(18,51)(19,50)(20,49)(21,48)(22,47)(23,46)(24,45)(25,44)(26,43)(27,42)(28,41)(29,40)(30,39)(31,38)(32,37)(33,36)(34,35)(65,152)(66,151)(67,150)(68,149)(69,148)(70,147)(71,146)(72,145)(73,144)(74,143)(75,142)(76,141)(77,140)(78,139)(79,138)(80,137)(81,136)(82,135)(83,134)(84,133)(85,132)(86,131)(87,130)(88,129)(89,128)(90,127)(91,126)(92,125)(93,124)(94,123)(95,122)(96,121)(97,120)(98,119)(99,118)(100,117)(101,116)(102,115)(103,114)(104,113)(105,112)(106,111)(107,110)(108,109)

Polycyclic presentation:

2^3,59,89|b^2=c,b^a=bc,d^a=d^58,e^a=e^88

Conjugacy classes & character table

10505 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, 2, 59, 89]);;
gap> SetPower(coll, 2, F.3);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 4, 1, F.4^58);;
gap> SetConjugate(coll, 5, 1, F.5^88);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D21004 #72274802