Dic₈₆₇₀ #72273008

order	34680
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₈₆₇₀

Splittings

Hall	C₃×C₅×C₂₈₉ ⋊ Q₈
Hall	C₅×C₂₈₉ ⋊ Dic₆
Hall	C₃×C₂₈₉ ⋊ Dic₁₀
Sylow	C₂₈₉ ⋊ Dic₃₀
Hall	C₃×C₅ ⋊ Dic₅₇₈
Sylow	C₅ ⋊ Dic₁₇₃₄
Sylow	C₃ ⋊ Dic₂₈₉₀

Subgroups

	subgroup	G/H
center	C₂	C₂×D₄₃₃₅
commutator, cosocle	C₂×C₃×C₅×C₂₈₉	C₂²
Frattini	C₂×C₁₇	C₂×D₂₅₅
Fitting	C₃×C₄×C₅×C₂₈₉	C₂
socle, prime radical	C₂×C₃×C₅×C₁₇	C₂×D₁₇
hypercenter	C₄	D₄₃₃₅
hypocenter	C₃×C₅×C₂₈₉	Q₈
LCS	G ▷ C₂×C₃×C₅×C₂₈₉ ▷ C₃×C₅×C₂₈₉
UCS	C₁ ◁ C₂ ◁ C₄

72 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 305
transitive degree	μ_t(G) = 34680

A minimal faithful permutation representation — 2 generators on 305 points:

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

Polycyclic presentation:

2^3,3,5,17^2|a^2=c,b^2=c,f^17=g,b^a=bc,d^a=d^2,e^a=e^4,f^a=f^16g^16,g^a=g^16

Conjugacy classes & character table

8673 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, 3, 5, 17, 17]);;
gap> SetPower(coll, 1, F.3);;
gap> SetPower(coll, 2, F.3);;
gap> SetPower(coll, 6, F.7);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 4, 1, F.4^2);;
gap> SetConjugate(coll, 5, 1, F.5^4);;
gap> SetConjugate(coll, 6, 1, F.6^16*F.7^16);;
gap> SetConjugate(coll, 7, 1, F.7^16);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic8670 #72273008