Dic₁₁₉₀₈ #72279415

order	47632
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₁₉₀₈

Splittings

Hall	C₁₃×C₂₂₉ ⋊ Q₁₆
Sylow	C₂₂₉ ⋊ Dic₅₂
Sylow	C₁₃ ⋊ Dic₉₁₆

Subgroups

	subgroup	G/H
center	C₂	D₁₁₉₀₈
commutator, cosocle	C₄×C₁₃×C₂₂₉	C₂²
Frattini	C₄	C₂×D₂₉₇₇
Fitting	C₈×C₁₃×C₂₂₉	C₂
socle, prime radical	C₂×C₁₃×C₂₂₉	D₄
hypercenter	C₈	D₂₉₇₇
hypocenter	C₁₃×C₂₂₉	Q₁₆
LCS	G ▷ C₄×C₁₃×C₂₂₉ ▷ C₂×C₁₃×C₂₂₉ ▷ C₁₃×C₂₂₉
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

36 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 258
transitive degree	μ_t(G) = 47632

A minimal faithful permutation representation — 2 generators on 258 points:

(1,2,3,4,5,6,7,8)(9,16,15,14,13,12,11,10)(17,18,19,20,21,22,23,24,25,26,27,28,29)(30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258);(1,9,5,13)(2,10,6,14)(3,11,7,15)(4,12,8,16)(18,29)(19,28)(20,27)(21,26)(22,25)(23,24)(31,258)(32,257)(33,256)(34,255)(35,254)(36,253)(37,252)(38,251)(39,250)(40,249)(41,248)(42,247)(43,246)(44,245)(45,244)(46,243)(47,242)(48,241)(49,240)(50,239)(51,238)(52,237)(53,236)(54,235)(55,234)(56,233)(57,232)(58,231)(59,230)(60,229)(61,228)(62,227)(63,226)(64,225)(65,224)(66,223)(67,222)(68,221)(69,220)(70,219)(71,218)(72,217)(73,216)(74,215)(75,214)(76,213)(77,212)(78,211)(79,210)(80,209)(81,208)(82,207)(83,206)(84,205)(85,204)(86,203)(87,202)(88,201)(89,200)(90,199)(91,198)(92,197)(93,196)(94,195)(95,194)(96,193)(97,192)(98,191)(99,190)(100,189)(101,188)(102,187)(103,186)(104,185)(105,184)(106,183)(107,182)(108,181)(109,180)(110,179)(111,178)(112,177)(113,176)(114,175)(115,174)(116,173)(117,172)(118,171)(119,170)(120,169)(121,168)(122,167)(123,166)(124,165)(125,164)(126,163)(127,162)(128,161)(129,160)(130,159)(131,158)(132,157)(133,156)(134,155)(135,154)(136,153)(137,152)(138,151)(139,150)(140,149)(141,148)(142,147)(143,146)(144,145)

Polycyclic presentation:

2^4,13,229|a^2=d,b^2=cd,c^2=d,b^a=bc,c^a=cd,e^a=e^12,f^a=f^228

Conjugacy classes & character table

11911 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, 2, 13, 229]);;
gap> SetPower(coll, 1, F.4);;
gap> SetPower(coll, 2, F.3*F.4);;
gap> SetPower(coll, 3, F.4);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 5, 1, F.5^12);;
gap> SetConjugate(coll, 6, 1, F.6^228);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic11908 #72279415