D₁₅₃₂₀ #72272044

order	30640
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₁₅₃₂₀

Splittings

Fitting	C₅×C₈×C₃₈₃ ⋊ C₂
Hall	C₅×C₃₈₃ ⋊ D₈
Sylow	C₃₈₃ ⋊ D₄₀
Sylow	C₅ ⋊ D₃₀₆₄

Subgroups

	subgroup	G/H
center	C₂	D₇₆₆₀
commutator, cosocle	C₄×C₅×C₃₈₃	C₂²
Frattini	C₄	C₂×D₁₉₁₅
Fitting	C₅×C₈×C₃₈₃	C₂
socle	C₂×C₅×C₃₈₃	D₄
hypercenter	C₈	D₁₉₁₅
hypocenter	C₅×C₃₈₃	D₈
LCS	G ▷ C₄×C₅×C₃₈₃ ▷ C₂×C₅×C₃₈₃ ▷ C₅×C₃₈₃
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

44 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 396
transitive degree	μ_t(G) = 15320

A minimal faithful permutation representation — 2 generators on 396 points:

(1,2,3,4,5,6,7,8)(9,10,11,12,13)(14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396);(2,8)(3,7)(4,6)(10,13)(11,12)(15,396)(16,395)(17,394)(18,393)(19,392)(20,391)(21,390)(22,389)(23,388)(24,387)(25,386)(26,385)(27,384)(28,383)(29,382)(30,381)(31,380)(32,379)(33,378)(34,377)(35,376)(36,375)(37,374)(38,373)(39,372)(40,371)(41,370)(42,369)(43,368)(44,367)(45,366)(46,365)(47,364)(48,363)(49,362)(50,361)(51,360)(52,359)(53,358)(54,357)(55,356)(56,355)(57,354)(58,353)(59,352)(60,351)(61,350)(62,349)(63,348)(64,347)(65,346)(66,345)(67,344)(68,343)(69,342)(70,341)(71,340)(72,339)(73,338)(74,337)(75,336)(76,335)(77,334)(78,333)(79,332)(80,331)(81,330)(82,329)(83,328)(84,327)(85,326)(86,325)(87,324)(88,323)(89,322)(90,321)(91,320)(92,319)(93,318)(94,317)(95,316)(96,315)(97,314)(98,313)(99,312)(100,311)(101,310)(102,309)(103,308)(104,307)(105,306)(106,305)(107,304)(108,303)(109,302)(110,301)(111,300)(112,299)(113,298)(114,297)(115,296)(116,295)(117,294)(118,293)(119,292)(120,291)(121,290)(122,289)(123,288)(124,287)(125,286)(126,285)(127,284)(128,283)(129,282)(130,281)(131,280)(132,279)(133,278)(134,277)(135,276)(136,275)(137,274)(138,273)(139,272)(140,271)(141,270)(142,269)(143,268)(144,267)(145,266)(146,265)(147,264)(148,263)(149,262)(150,261)(151,260)(152,259)(153,258)(154,257)(155,256)(156,255)(157,254)(158,253)(159,252)(160,251)(161,250)(162,249)(163,248)(164,247)(165,246)(166,245)(167,244)(168,243)(169,242)(170,241)(171,240)(172,239)(173,238)(174,237)(175,236)(176,235)(177,234)(178,233)(179,232)(180,231)(181,230)(182,229)(183,228)(184,227)(185,226)(186,225)(187,224)(188,223)(189,222)(190,221)(191,220)(192,219)(193,218)(194,217)(195,216)(196,215)(197,214)(198,213)(199,212)(200,211)(201,210)(202,209)(203,208)(204,207)(205,206)

Polycyclic presentation:

2^4,5,383|b^2=cd,c^2=d,b^a=bc,c^a=cd,e^a=e^4,f^a=f^382

Conjugacy classes & character table

7663 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₃₀₆₄₀
maximal normal subgroup of	D₃₀₆₄₀
maximal quotient of	Dic₁₅₃₂₀, D₃₀₆₄₀

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, 383]);;
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.6^382);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D15320 #72272044