Dic₄₅₄₀ #72216016

order	18160
automorphism	Aut(G) order 32833280
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₄₅₄₀

Splittings

Hall	C₅×C₂₂₇ ⋊ Q₁₆
Sylow	C₂₂₇ ⋊ Dic₂₀
Sylow	C₅ ⋊ Dic₉₀₈

Subgroups

	subgroup	G/H
center	C₂	D₄₅₄₀
commutator, cosocle	C₄×C₅×C₂₂₇	C₂²
Frattini	C₄	C₂×D₁₁₃₅
Fitting	C₅×C₈×C₂₂₇	C₂
socle, prime radical	C₂×C₅×C₂₂₇	D₄
hypercenter	C₈	D₁₁₃₅
hypocenter	C₅×C₂₂₇	Q₁₆
LCS	G ▷ C₄×C₅×C₂₂₇ ▷ C₂×C₅×C₂₂₇ ▷ C₅×C₂₂₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

36 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 248
transitive degree	μ_t(G) = 18160

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

Polycyclic presentation:

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

Conjugacy classes & character table

4543 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₉₀₈₀, Dic₁₃₆₂₀
maximal normal subgroup of	Dic₉₀₈₀
maximal quotient of	Dic₁₃₆₂₀

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, 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.5^4);;
gap> SetConjugate(coll, 6, 1, F.6^226);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic4540 #72216016