Dic₁₃₆₂₀ #72289772

order	54480
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₃₆₂₀

Splittings

Hall	C₃×C₅×C₂₂₇ ⋊ Q₁₆
Hall	C₅×C₂₂₇ ⋊ Dic₁₂
Hall	C₃×C₂₂₇ ⋊ Dic₂₀
Sylow	C₂₂₇ ⋊ Dic₆₀
Hall	C₃×C₅ ⋊ Dic₉₀₈
Sylow	C₅ ⋊ Dic₂₇₂₄
Sylow	C₃ ⋊ Dic₄₅₄₀

Subgroups

	subgroup	G/H
center	C₂	D₁₃₆₂₀
commutator, cosocle	C₃×C₄×C₅×C₂₂₇	C₂²
Frattini	C₄	C₂×D₃₄₀₅
Fitting	C₃×C₅×C₈×C₂₂₇	C₂
socle, prime radical	C₂×C₃×C₅×C₂₂₇	D₄
hypercenter	C₈	D₃₄₀₅
hypocenter	C₃×C₅×C₂₂₇	Q₁₆
LCS	G ▷ C₃×C₄×C₅×C₂₂₇ ▷ C₂×C₃×C₅×C₂₂₇ ▷ C₃×C₅×C₂₂₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

72 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 251
transitive degree	μ_t(G) = 54480

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

Polycyclic presentation:

2^4,3,5,227|a^2=d,b^2=cd,c^2=d,b^a=bc,c^a=cd,e^a=e^2,f^a=f^4,g^a=g^226

Conjugacy classes & character table

13623 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", "g");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 3, 5, 227]);;
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.5^2);;
gap> SetConjugate(coll, 6, 1, F.6^4);;
gap> SetConjugate(coll, 7, 1, F.7^226);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic13620 #72289772