Dic₃₆₂₀ #72214647

order	14480
automorphism	Aut(G) order 20851200
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₃₆₂₀

Splittings

Hall	C₅×C₁₈₁ ⋊ Q₁₆
Sylow	C₁₈₁ ⋊ Dic₂₀
Sylow	C₅ ⋊ Dic₇₂₄

Subgroups

	subgroup	G/H
center	C₂	D₃₆₂₀
commutator, cosocle	C₄×C₅×C₁₈₁	C₂²
Frattini	C₄	C₂×D₉₀₅
Fitting	C₅×C₈×C₁₈₁	C₂
socle, prime radical	C₂×C₅×C₁₈₁	D₄
hypercenter	C₈	D₉₀₅
hypocenter	C₅×C₁₈₁	Q₁₆
LCS	G ▷ C₄×C₅×C₁₈₁ ▷ C₂×C₅×C₁₈₁ ▷ C₅×C₁₈₁
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

36 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 202
transitive degree	μ_t(G) = 14480

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

Polycyclic presentation:

2^4,5,181|a^2=d,b^2=cd,c^2=d,b^a=bc,c^a=cd,e^a=e^4,f^a=f^180

Conjugacy classes & character table

3623 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₇₂₄₀, Dic₁₀₈₆₀
maximal normal subgroup of	Dic₇₂₄₀
maximal quotient of	Dic₁₀₈₆₀

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, 181]);;
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.6^180);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic3620 #72214647