C₅×C₃⋊(D₃₆₁⋊C₃) #44570438

order	32490
automorphism	?
homology	Abelianization (H₁) = C₂×C₃×C₅ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₅×C₃⋊(D₃₆₁⋊C₃)
unambiguous semidirect product	C₃⋊(C₅×D₃₆₁⋊C₃)
unambiguous semidirect product	(C₃₆₁⋊C₃)⋊(C₅×D₃)
unambiguous semidirect product	D₁₀₈₃⋊(C₃×C₅)
unambiguous semidirect product	(C₅×D₁₀₈₃)⋊C₃

Splittings

Fitting	C₃×C₅×C₃₆₁ ⋊ C₂×C₃
derived	C₃×C₃₆₁ ⋊ C₂×C₃×C₅
Hall	C₃×C₅×C₃₆₁⋊C₃ ⋊ C₂
Hall	C₃×C₃₆₁⋊C₃ ⋊ C₂×C₅
Hall	C₅×C₃₆₁ ⋊ C₃×D₃
Sylow	C₃₆₁ ⋊ C₃×C₅×D₃

Subgroups

	subgroup	G/H
center, hypercenter	C₅	C₃⋊(D₃₆₁⋊C₃)
commutator, cosocle, hypocenter	C₃×C₃₆₁	C₂×C₃×C₅
Frattini	C₁₉	C₅×C₃⋊(D₁₉⋊C₃)
Fitting	C₃×C₅×C₃₆₁	C₂×C₃
socle	C₃×C₅×C₁₉	D₁₉⋊C₃
LCS	G ▷ C₃×C₃₆₁
UCS	C₁ ◁ C₅

54 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 369
transitive degree	μ_t(G) = 5415

A minimal faithful permutation representation — 2 generators on 369 points:

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

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(32490, 22);

C5×C3⋊(D361⋊C3) #44570438