Dic₁₄₈₄₂ #72292136

order	59368
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₄₈₄₂

Splittings

Hall	C₄₁×C₁₈₁ ⋊ Q₈
Sylow	C₁₈₁ ⋊ Dic₈₂
Sylow	C₄₁ ⋊ Dic₃₆₂

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₇₄₂₁
commutator, socle, prime radical, cosocle	C₂×C₄₁×C₁₈₁	C₂²
Fitting	C₄×C₄₁×C₁₈₁	C₂
hypercenter	C₄	D₇₄₂₁
hypocenter	C₄₁×C₁₈₁	Q₈
LCS	G ▷ C₂×C₄₁×C₁₈₁ ▷ C₄₁×C₁₈₁
UCS	C₁ ◁ C₂ ◁ C₄

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 230
transitive degree	μ_t(G) = 59368

A minimal faithful permutation representation — 2 generators on 230 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)(50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230);(1,5,3,7)(2,6,4,8)(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)(51,230)(52,229)(53,228)(54,227)(55,226)(56,225)(57,224)(58,223)(59,222)(60,221)(61,220)(62,219)(63,218)(64,217)(65,216)(66,215)(67,214)(68,213)(69,212)(70,211)(71,210)(72,209)(73,208)(74,207)(75,206)(76,205)(77,204)(78,203)(79,202)(80,201)(81,200)(82,199)(83,198)(84,197)(85,196)(86,195)(87,194)(88,193)(89,192)(90,191)(91,190)(92,189)(93,188)(94,187)(95,186)(96,185)(97,184)(98,183)(99,182)(100,181)(101,180)(102,179)(103,178)(104,177)(105,176)(106,175)(107,174)(108,173)(109,172)(110,171)(111,170)(112,169)(113,168)(114,167)(115,166)(116,165)(117,164)(118,163)(119,162)(120,161)(121,160)(122,159)(123,158)(124,157)(125,156)(126,155)(127,154)(128,153)(129,152)(130,151)(131,150)(132,149)(133,148)(134,147)(135,146)(136,145)(137,144)(138,143)(139,142)(140,141)

Polycyclic presentation:

2^3,41,181|a^2=c,b^2=c,b^a=bc,d^a=d^40,e^a=e^180

Conjugacy classes & character table

14845 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, 41, 181]);;
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.4^40);;
gap> SetConjugate(coll, 5, 1, F.5^180);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic14842 #72292136