Dic₁₂₆₆ #72211844

order	5064
automorphism	Aut(G) order 2126880
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₂₆₆

Splittings

Hall	C₃×C₂₁₁ ⋊ Q₈
Sylow	C₂₁₁ ⋊ Dic₆
Sylow	C₃ ⋊ Dic₄₂₂

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₆₃₃
commutator, socle, prime radical, cosocle	C₂×C₃×C₂₁₁	C₂²
Fitting	C₃×C₄×C₂₁₁	C₂
hypercenter	C₄	D₆₃₃
hypocenter	C₃×C₂₁₁	Q₈
LCS	G ▷ C₂×C₃×C₂₁₁ ▷ C₃×C₂₁₁
UCS	C₁ ◁ C₂ ◁ C₄

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 222
transitive degree	μ_t(G) = 5064

A minimal faithful permutation representation — 2 generators on 222 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);(1,5,3,7)(2,6,4,8)(10,11)(13,222)(14,221)(15,220)(16,219)(17,218)(18,217)(19,216)(20,215)(21,214)(22,213)(23,212)(24,211)(25,210)(26,209)(27,208)(28,207)(29,206)(30,205)(31,204)(32,203)(33,202)(34,201)(35,200)(36,199)(37,198)(38,197)(39,196)(40,195)(41,194)(42,193)(43,192)(44,191)(45,190)(46,189)(47,188)(48,187)(49,186)(50,185)(51,184)(52,183)(53,182)(54,181)(55,180)(56,179)(57,178)(58,177)(59,176)(60,175)(61,174)(62,173)(63,172)(64,171)(65,170)(66,169)(67,168)(68,167)(69,166)(70,165)(71,164)(72,163)(73,162)(74,161)(75,160)(76,159)(77,158)(78,157)(79,156)(80,155)(81,154)(82,153)(83,152)(84,151)(85,150)(86,149)(87,148)(88,147)(89,146)(90,145)(91,144)(92,143)(93,142)(94,141)(95,140)(96,139)(97,138)(98,137)(99,136)(100,135)(101,134)(102,133)(103,132)(104,131)(105,130)(106,129)(107,128)(108,127)(109,126)(110,125)(111,124)(112,123)(113,122)(114,121)(115,120)(116,119)(117,118)

Polycyclic presentation:

2^3,3,211|a^2=c,b^2=c,b^a=bc,d^a=d^2,e^a=e^210

Conjugacy classes & character table

1269 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₂₅₃₂, Dic₃₇₉₈, Dic₆₃₃₀, …
maximal normal subgroup of	Dic₂₅₃₂
maximal quotient of	Dic₃₇₉₈, Dic₆₃₃₀, Dic₈₈₆₂, …

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, 3, 211]);;
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.5^210);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic1266 #72211844