C₅×C₃⋊D₁₀₈₃ #44570450

order	32490
automorphism	?
homology	Abelianization (H₁) = C₂×C₅ · Schur multiplier (H₂) = C₃

Aliases

direct product	C₅×C₃⋊D₁₀₈₃
unambiguous semidirect product	C₃⋊(C₅×D₁₀₈₃)
unambiguous semidirect product	C₃₆₁⋊(C₅×C₃⋊D₃)

Splittings

Fitting, Hall	C₃²×C₅×C₃₆₁ ⋊ C₂
derived, Hall	C₃²×C₃₆₁ ⋊ C₂×C₅
Hall	C₅×C₃₆₁ ⋊ C₃⋊D₃
Sylow	C₃₆₁ ⋊ C₅×C₃⋊D₃
Hall	C₃²×C₅ ⋊ D₃₆₁
Sylow	C₃² ⋊ C₅×D₃₆₁

Subgroups

	subgroup	G/H
center, hypercenter	C₅	C₃⋊D₁₀₈₃
commutator, cosocle, hypocenter	C₃²×C₃₆₁	C₂×C₅
Frattini	C₁₉	C₅×C₃⋊D₅₇
Fitting	C₃²×C₅×C₃₆₁	C₂
socle	C₃²×C₅×C₁₉	D₁₉
LCS	G ▷ C₃²×C₃₆₁
UCS	C₁ ◁ C₅

72 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 372
transitive degree	μ_t(G) = 16245

A minimal faithful permutation representation — 3 generators on 372 points:

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

Conjugacy classes & character table

8130 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(32490, 34);

C5×C3⋊D1083 #44570450