Dic₁₅₀₆₄ #72292491

order	60256
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₅₀₆₄

Splittings

Hall	C₇×C₂₆₉ ⋊ Q₃₂
Sylow	C₂₆₉ ⋊ Dic₅₆
Sylow	C₇ ⋊ Dic₂₁₅₂

Subgroups

	subgroup	G/H
center	C₂	D₁₅₀₆₄
commutator, cosocle	C₇×C₈×C₂₆₉	C₂²
Frattini	C₈	C₂×D₁₈₈₃
Fitting	C₇×C₁₆×C₂₆₉	C₂
socle, prime radical	C₂×C₇×C₂₆₉	D₈
hypercenter	C₁₆	D₁₈₈₃
hypocenter	C₇×C₂₆₉	Q₃₂
LCS	G ▷ C₇×C₈×C₂₆₉ ▷ C₄×C₇×C₂₆₉ ▷ C₂×C₇×C₂₆₉ ▷ C₇×C₂₆₉
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆

48 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 308
transitive degree	μ_t(G) = 60256

A minimal faithful permutation representation — 2 generators on 308 points:

(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16)(17,32,31,30,29,28,27,26,25,24,23,22,21,20,19,18)(33,34,35,36,37,38,39)(40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308);(1,17,9,25)(2,18,10,26)(3,19,11,27)(4,20,12,28)(5,21,13,29)(6,22,14,30)(7,23,15,31)(8,24,16,32)(34,39)(35,38)(36,37)(41,308)(42,307)(43,306)(44,305)(45,304)(46,303)(47,302)(48,301)(49,300)(50,299)(51,298)(52,297)(53,296)(54,295)(55,294)(56,293)(57,292)(58,291)(59,290)(60,289)(61,288)(62,287)(63,286)(64,285)(65,284)(66,283)(67,282)(68,281)(69,280)(70,279)(71,278)(72,277)(73,276)(74,275)(75,274)(76,273)(77,272)(78,271)(79,270)(80,269)(81,268)(82,267)(83,266)(84,265)(85,264)(86,263)(87,262)(88,261)(89,260)(90,259)(91,258)(92,257)(93,256)(94,255)(95,254)(96,253)(97,252)(98,251)(99,250)(100,249)(101,248)(102,247)(103,246)(104,245)(105,244)(106,243)(107,242)(108,241)(109,240)(110,239)(111,238)(112,237)(113,236)(114,235)(115,234)(116,233)(117,232)(118,231)(119,230)(120,229)(121,228)(122,227)(123,226)(124,225)(125,224)(126,223)(127,222)(128,221)(129,220)(130,219)(131,218)(132,217)(133,216)(134,215)(135,214)(136,213)(137,212)(138,211)(139,210)(140,209)(141,208)(142,207)(143,206)(144,205)(145,204)(146,203)(147,202)(148,201)(149,200)(150,199)(151,198)(152,197)(153,196)(154,195)(155,194)(156,193)(157,192)(158,191)(159,190)(160,189)(161,188)(162,187)(163,186)(164,185)(165,184)(166,183)(167,182)(168,181)(169,180)(170,179)(171,178)(172,177)(173,176)(174,175)

Polycyclic presentation:

2^5,7,269|a^2=e,b^2=cd,c^2=de,d^2=e,b^a=bc,c^a=cd,d^a=de,f^a=f^6,g^a=g^268

Conjugacy classes & character table

15067 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, 2, 7, 269]);;
gap> SetPower(coll, 1, F.5);;
gap> SetPower(coll, 2, F.3*F.4);;
gap> SetPower(coll, 3, F.4*F.5);;
gap> SetPower(coll, 4, F.5);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 4, 1, F.4*F.5);;
gap> SetConjugate(coll, 6, 1, F.6^6);;
gap> SetConjugate(coll, 7, 1, F.7^268);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic15064 #72292491