C₅×C₃⋊(D₃₄₉⋊C₃) #44536115

order	31410
automorphism	?
homology	Abelianization (H₁) = C₂×C₃×C₅ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₅×C₃⋊(D₃₄₉⋊C₃)
unambiguous semidirect product	C₃⋊(C₅×D₃₄₉⋊C₃)
unambiguous semidirect product	(C₃₄₉⋊C₃)⋊(C₅×D₃)
unambiguous semidirect product	D₁₀₄₇⋊(C₃×C₅)
unambiguous semidirect product	(C₅×D₁₀₄₇)⋊C₃

Splittings

Fitting, socle	C₃×C₅×C₃₄₉ ⋊ C₂×C₃
derived	C₃×C₃₄₉ ⋊ C₂×C₃×C₅
Hall	C₃×C₅×C₃₄₉⋊C₃ ⋊ C₂
Hall	C₃×C₃₄₉⋊C₃ ⋊ C₂×C₅
Hall	C₅×C₃₄₉ ⋊ C₃×D₃
Sylow	C₃₄₉ ⋊ C₃×C₅×D₃

Subgroups

	subgroup	G/H
center, hypercenter	C₅	C₃⋊(D₃₄₉⋊C₃)
commutator, cosocle, hypocenter	C₃×C₃₄₉	C₂×C₃×C₅
Frattini	C₁	G
Fitting, socle	C₃×C₅×C₃₄₉	C₂×C₃
LCS	G ▷ C₃×C₃₄₉
UCS	C₁ ◁ C₅

36 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 357
transitive degree	μ_t(G) = 5235

A minimal faithful permutation representation — 2 generators on 357 points:

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

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(31410, 17);

C5×C3⋊(D349⋊C3) #44536115