Dic₁₅₀₃₈ #72292449

order	60152
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₅₀₃₈

Splittings

Hall	C₇₃×C₁₀₃ ⋊ Q₈
Sylow	C₁₀₃ ⋊ Dic₁₄₆
Sylow	C₇₃ ⋊ Dic₂₀₆

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₇₅₁₉
commutator, socle, prime radical, cosocle	C₂×C₇₃×C₁₀₃	C₂²
Fitting	C₄×C₇₃×C₁₀₃	C₂
hypercenter	C₄	D₇₅₁₉
hypocenter	C₇₃×C₁₀₃	Q₈
LCS	G ▷ C₂×C₇₃×C₁₀₃ ▷ C₇₃×C₁₀₃
UCS	C₁ ◁ C₂ ◁ C₄

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 184
transitive degree	μ_t(G) = 60152

A minimal faithful permutation representation — 2 generators on 184 points:

(1,2,3,4)(5,8,7,6)(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)(82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184);(1,5,3,7)(2,6,4,8)(10,81)(11,80)(12,79)(13,78)(14,77)(15,76)(16,75)(17,74)(18,73)(19,72)(20,71)(21,70)(22,69)(23,68)(24,67)(25,66)(26,65)(27,64)(28,63)(29,62)(30,61)(31,60)(32,59)(33,58)(34,57)(35,56)(36,55)(37,54)(38,53)(39,52)(40,51)(41,50)(42,49)(43,48)(44,47)(45,46)(83,184)(84,183)(85,182)(86,181)(87,180)(88,179)(89,178)(90,177)(91,176)(92,175)(93,174)(94,173)(95,172)(96,171)(97,170)(98,169)(99,168)(100,167)(101,166)(102,165)(103,164)(104,163)(105,162)(106,161)(107,160)(108,159)(109,158)(110,157)(111,156)(112,155)(113,154)(114,153)(115,152)(116,151)(117,150)(118,149)(119,148)(120,147)(121,146)(122,145)(123,144)(124,143)(125,142)(126,141)(127,140)(128,139)(129,138)(130,137)(131,136)(132,135)(133,134)

Polycyclic presentation:

2^3,73,103|a^2=c,b^2=c,b^a=bc,d^a=d^72,e^a=e^102

Conjugacy classes & character table

15041 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, 73, 103]);;
gap> SetPower(coll, 1, F.3);;
gap> SetPower(coll, 2, F.3);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 4, 1, F.4^72);;
gap> SetConjugate(coll, 5, 1, F.5^102);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic15038 #72292449