Dic₃₁₇₀ #72213481

order	12680
automorphism	Aut(G) order 16027520
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₃₁₇₀

Splittings

Hall	C₅×C₃₁₇ ⋊ Q₈
Sylow	C₃₁₇ ⋊ Dic₁₀
Sylow	C₅ ⋊ Dic₆₃₄

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₁₅₈₅
commutator, socle, prime radical, cosocle	C₂×C₅×C₃₁₇	C₂²
Fitting	C₄×C₅×C₃₁₇	C₂
hypercenter	C₄	D₁₅₈₅
hypocenter	C₅×C₃₁₇	Q₈
LCS	G ▷ C₂×C₅×C₃₁₇ ▷ C₅×C₃₁₇
UCS	C₁ ◁ C₂ ◁ C₄

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 330
transitive degree	μ_t(G) = 12680

A minimal faithful permutation representation — 2 generators on 330 points:

(1,2,3,4)(5,8,7,6)(9,10,11,12,13)(14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330);(1,5,3,7)(2,6,4,8)(10,13)(11,12)(15,330)(16,329)(17,328)(18,327)(19,326)(20,325)(21,324)(22,323)(23,322)(24,321)(25,320)(26,319)(27,318)(28,317)(29,316)(30,315)(31,314)(32,313)(33,312)(34,311)(35,310)(36,309)(37,308)(38,307)(39,306)(40,305)(41,304)(42,303)(43,302)(44,301)(45,300)(46,299)(47,298)(48,297)(49,296)(50,295)(51,294)(52,293)(53,292)(54,291)(55,290)(56,289)(57,288)(58,287)(59,286)(60,285)(61,284)(62,283)(63,282)(64,281)(65,280)(66,279)(67,278)(68,277)(69,276)(70,275)(71,274)(72,273)(73,272)(74,271)(75,270)(76,269)(77,268)(78,267)(79,266)(80,265)(81,264)(82,263)(83,262)(84,261)(85,260)(86,259)(87,258)(88,257)(89,256)(90,255)(91,254)(92,253)(93,252)(94,251)(95,250)(96,249)(97,248)(98,247)(99,246)(100,245)(101,244)(102,243)(103,242)(104,241)(105,240)(106,239)(107,238)(108,237)(109,236)(110,235)(111,234)(112,233)(113,232)(114,231)(115,230)(116,229)(117,228)(118,227)(119,226)(120,225)(121,224)(122,223)(123,222)(124,221)(125,220)(126,219)(127,218)(128,217)(129,216)(130,215)(131,214)(132,213)(133,212)(134,211)(135,210)(136,209)(137,208)(138,207)(139,206)(140,205)(141,204)(142,203)(143,202)(144,201)(145,200)(146,199)(147,198)(148,197)(149,196)(150,195)(151,194)(152,193)(153,192)(154,191)(155,190)(156,189)(157,188)(158,187)(159,186)(160,185)(161,184)(162,183)(163,182)(164,181)(165,180)(166,179)(167,178)(168,177)(169,176)(170,175)(171,174)(172,173)

Polycyclic presentation:

2^3,5,317|a^2=c,b^2=c,b^a=bc,d^a=d^4,e^a=e^316

Conjugacy classes & character table

3173 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₆₃₄₀, Dic₉₅₁₀, Dic₁₅₈₅₀
maximal normal subgroup of	Dic₆₃₄₀
maximal quotient of	Dic₉₅₁₀, Dic₁₅₈₅₀

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e");;
gap> coll := SingleCollector(F, [2, 2, 2, 5, 317]);;
gap> SetPower(coll, 1, F.3);;
gap> SetPower(coll, 2, F.3);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 4, 1, F.4^4);;
gap> SetConjugate(coll, 5, 1, F.5^316);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic3170 #72213481