Dic₁₁₉₄₆ #72279727

order	47784
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₁₉₄₆

Splittings

Hall	C₃×C₁₁×C₁₈₁ ⋊ Q₈
Hall	C₁₁×C₁₈₁ ⋊ Dic₆
Hall	C₃×C₁₈₁ ⋊ Dic₂₂
Sylow	C₁₈₁ ⋊ Dic₆₆
Hall	C₃×C₁₁ ⋊ Dic₃₆₂
Sylow	C₁₁ ⋊ Dic₁₀₈₆
Sylow	C₃ ⋊ Dic₃₉₈₂

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₅₉₇₃
commutator, socle, prime radical, cosocle	C₂×C₃×C₁₁×C₁₈₁	C₂²
Fitting	C₃×C₄×C₁₁×C₁₈₁	C₂
hypercenter	C₄	D₅₉₇₃
hypocenter	C₃×C₁₁×C₁₈₁	Q₈
LCS	G ▷ C₂×C₃×C₁₁×C₁₈₁ ▷ C₃×C₁₁×C₁₈₁
UCS	C₁ ◁ C₂ ◁ C₄

48 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 203
transitive degree	μ_t(G) = 47784

A minimal faithful permutation representation — 2 generators on 203 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);(1,5,3,7)(2,6,4,8)(10,11)(13,22)(14,21)(15,20)(16,19)(17,18)(24,203)(25,202)(26,201)(27,200)(28,199)(29,198)(30,197)(31,196)(32,195)(33,194)(34,193)(35,192)(36,191)(37,190)(38,189)(39,188)(40,187)(41,186)(42,185)(43,184)(44,183)(45,182)(46,181)(47,180)(48,179)(49,178)(50,177)(51,176)(52,175)(53,174)(54,173)(55,172)(56,171)(57,170)(58,169)(59,168)(60,167)(61,166)(62,165)(63,164)(64,163)(65,162)(66,161)(67,160)(68,159)(69,158)(70,157)(71,156)(72,155)(73,154)(74,153)(75,152)(76,151)(77,150)(78,149)(79,148)(80,147)(81,146)(82,145)(83,144)(84,143)(85,142)(86,141)(87,140)(88,139)(89,138)(90,137)(91,136)(92,135)(93,134)(94,133)(95,132)(96,131)(97,130)(98,129)(99,128)(100,127)(101,126)(102,125)(103,124)(104,123)(105,122)(106,121)(107,120)(108,119)(109,118)(110,117)(111,116)(112,115)(113,114)

Polycyclic presentation:

2^3,3,11,181|a^2=c,b^2=c,b^a=bc,d^a=d^2,e^a=e^10,f^a=f^180

Conjugacy classes & character table

11949 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", "f");;
gap> coll := SingleCollector(F, [2, 2, 2, 3, 11, 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.4^2);;
gap> SetConjugate(coll, 5, 1, F.5^10);;
gap> SetConjugate(coll, 6, 1, F.6^180);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic11946 #72279727