Dic₁₃₃₉₄ #72289199

order	53576
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₃₃₉₄

Splittings

Hall	C₃₇×C₁₈₁ ⋊ Q₈
Sylow	C₁₈₁ ⋊ Dic₇₄
Sylow	C₃₇ ⋊ Dic₃₆₂

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₆₆₉₇
commutator, socle, prime radical, cosocle	C₂×C₃₇×C₁₈₁	C₂²
Fitting	C₄×C₃₇×C₁₈₁	C₂
hypercenter	C₄	D₆₆₉₇
hypocenter	C₃₇×C₁₈₁	Q₈
LCS	G ▷ C₂×C₃₇×C₁₈₁ ▷ C₃₇×C₁₈₁
UCS	C₁ ◁ C₂ ◁ C₄

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 226
transitive degree	μ_t(G) = 53576

A minimal faithful permutation representation — 2 generators on 226 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);(1,5,3,7)(2,6,4,8)(10,45)(11,44)(12,43)(13,42)(14,41)(15,40)(16,39)(17,38)(18,37)(19,36)(20,35)(21,34)(22,33)(23,32)(24,31)(25,30)(26,29)(27,28)(47,226)(48,225)(49,224)(50,223)(51,222)(52,221)(53,220)(54,219)(55,218)(56,217)(57,216)(58,215)(59,214)(60,213)(61,212)(62,211)(63,210)(64,209)(65,208)(66,207)(67,206)(68,205)(69,204)(70,203)(71,202)(72,201)(73,200)(74,199)(75,198)(76,197)(77,196)(78,195)(79,194)(80,193)(81,192)(82,191)(83,190)(84,189)(85,188)(86,187)(87,186)(88,185)(89,184)(90,183)(91,182)(92,181)(93,180)(94,179)(95,178)(96,177)(97,176)(98,175)(99,174)(100,173)(101,172)(102,171)(103,170)(104,169)(105,168)(106,167)(107,166)(108,165)(109,164)(110,163)(111,162)(112,161)(113,160)(114,159)(115,158)(116,157)(117,156)(118,155)(119,154)(120,153)(121,152)(122,151)(123,150)(124,149)(125,148)(126,147)(127,146)(128,145)(129,144)(130,143)(131,142)(132,141)(133,140)(134,139)(135,138)(136,137)

Polycyclic presentation:

2^3,37,181|a^2=c,b^2=c,b^a=bc,d^a=d^36,e^a=e^180

Conjugacy classes & character table

13397 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, 37, 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.4^36);;
gap> SetConjugate(coll, 5, 1, F.5^180);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic13394 #72289199