Dic₁₆₃₀₀ #72301516

order	65200
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₆₃₀₀

Splittings

Hall	C₂₅×C₁₆₃ ⋊ Q₁₆
Sylow	C₁₆₃ ⋊ Dic₁₀₀
Sylow	C₂₅ ⋊ Dic₆₅₂

Subgroups

	subgroup	G/H
center	C₂	D₁₆₃₀₀
commutator, cosocle	C₄×C₂₅×C₁₆₃	C₂²
Frattini	C₄×C₅	C₂×D₈₁₅
Fitting	C₈×C₂₅×C₁₆₃	C₂
socle, prime radical	C₂×C₅×C₁₆₃	D₂₀
hypercenter	C₈	D₄₀₇₅
hypocenter	C₂₅×C₁₆₃	Q₁₆
LCS	G ▷ C₄×C₂₅×C₁₆₃ ▷ C₂×C₂₅×C₁₆₃ ▷ C₂₅×C₁₆₃
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

54 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 204
transitive degree	μ_t(G) = 65200

A minimal faithful permutation representation — 2 generators on 204 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);(1,9,5,13)(2,10,6,14)(3,11,7,15)(4,12,8,16)(18,41)(19,40)(20,39)(21,38)(22,37)(23,36)(24,35)(25,34)(26,33)(27,32)(28,31)(29,30)(43,204)(44,203)(45,202)(46,201)(47,200)(48,199)(49,198)(50,197)(51,196)(52,195)(53,194)(54,193)(55,192)(56,191)(57,190)(58,189)(59,188)(60,187)(61,186)(62,185)(63,184)(64,183)(65,182)(66,181)(67,180)(68,179)(69,178)(70,177)(71,176)(72,175)(73,174)(74,173)(75,172)(76,171)(77,170)(78,169)(79,168)(80,167)(81,166)(82,165)(83,164)(84,163)(85,162)(86,161)(87,160)(88,159)(89,158)(90,157)(91,156)(92,155)(93,154)(94,153)(95,152)(96,151)(97,150)(98,149)(99,148)(100,147)(101,146)(102,145)(103,144)(104,143)(105,142)(106,141)(107,140)(108,139)(109,138)(110,137)(111,136)(112,135)(113,134)(114,133)(115,132)(116,131)(117,130)(118,129)(119,128)(120,127)(121,126)(122,125)(123,124)

Polycyclic presentation:

2^4,5,163,5|a^2=d,b^2=cd,c^2=d,e^5=g,b^a=bc,c^a=cd,e^a=e^4g^4,f^a=f^162,g^a=g^4

Conjugacy classes & character table

16303 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, 5, 163, 5]);;
gap> SetPower(coll, 1, F.4);;
gap> SetPower(coll, 2, F.3*F.4);;
gap> SetPower(coll, 3, F.4);;
gap> SetPower(coll, 5, F.7);;
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*F.7^4);;
gap> SetConjugate(coll, 6, 1, F.6^162);;
gap> SetConjugate(coll, 7, 1, F.7^4);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic16300 #72301516