Dic₉₀₂₅ #49115443

order	36100
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₉₀₂₅
unambiguous semidirect product	C₃₆₁⋊Dic₂₅

Splittings

derived, Hall	C₂₅×C₃₆₁ ⋊ C₄
Sylow	C₃₆₁ ⋊ Dic₂₅
Sylow	C₂₅ ⋊ Dic₃₆₁

Subgroups

	subgroup	G/H
center, hypercenter	C₂	D₉₀₂₅
commutator, hypocenter	C₂₅×C₃₆₁	C₄
Frattini, socle, prime radical	C₂×C₅×C₁₉	D₉₅
Fitting, cosocle	C₂×C₂₅×C₃₆₁	C₂
LCS	G ▷ C₂₅×C₃₆₁
UCS	C₁ ◁ C₂

27 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 390
transitive degree	μ_t(G) = 36100

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

Polycyclic presentation:

2,5,19,2,5,19|a^2=d,b^5=e^4,c^19=f^18,b^a=b^4e,c^a=c^18f,e^a=e^4,f^a=f^18

Conjugacy classes & character table

9028 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(36100, 4);

Dic9025 #49115443