D₁₃₉₁₉ #42123770

order	27838
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₁₃₉₁₉
unambiguous semidirect product	C₃₁⋊D₄₄₉
unambiguous semidirect product	C₄₄₉⋊D₃₁

Splittings

Fitting, socle, derived, Hall	C₃₁×C₄₄₉ ⋊ C₂
Sylow	C₄₄₉ ⋊ D₃₁
Sylow	C₃₁ ⋊ D₄₄₉

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₃₁×C₄₄₉	C₂
LCS	G ▷ C₃₁×C₄₄₉
UCS	C₁

8 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 480
transitive degree	μ_t(G) = 13919

A minimal faithful permutation representation — 2 generators on 480 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)(32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456,457,458,459,460,461,462,463,464,465,466,467,468,469,470,471,472,473,474,475,476,477,478,479,480);(2,31)(3,30)(4,29)(5,28)(6,27)(7,26)(8,25)(9,24)(10,23)(11,22)(12,21)(13,20)(14,19)(15,18)(16,17)(33,480)(34,479)(35,478)(36,477)(37,476)(38,475)(39,474)(40,473)(41,472)(42,471)(43,470)(44,469)(45,468)(46,467)(47,466)(48,465)(49,464)(50,463)(51,462)(52,461)(53,460)(54,459)(55,458)(56,457)(57,456)(58,455)(59,454)(60,453)(61,452)(62,451)(63,450)(64,449)(65,448)(66,447)(67,446)(68,445)(69,444)(70,443)(71,442)(72,441)(73,440)(74,439)(75,438)(76,437)(77,436)(78,435)(79,434)(80,433)(81,432)(82,431)(83,430)(84,429)(85,428)(86,427)(87,426)(88,425)(89,424)(90,423)(91,422)(92,421)(93,420)(94,419)(95,418)(96,417)(97,416)(98,415)(99,414)(100,413)(101,412)(102,411)(103,410)(104,409)(105,408)(106,407)(107,406)(108,405)(109,404)(110,403)(111,402)(112,401)(113,400)(114,399)(115,398)(116,397)(117,396)(118,395)(119,394)(120,393)(121,392)(122,391)(123,390)(124,389)(125,388)(126,387)(127,386)(128,385)(129,384)(130,383)(131,382)(132,381)(133,380)(134,379)(135,378)(136,377)(137,376)(138,375)(139,374)(140,373)(141,372)(142,371)(143,370)(144,369)(145,368)(146,367)(147,366)(148,365)(149,364)(150,363)(151,362)(152,361)(153,360)(154,359)(155,358)(156,357)(157,356)(158,355)(159,354)(160,353)(161,352)(162,351)(163,350)(164,349)(165,348)(166,347)(167,346)(168,345)(169,344)(170,343)(171,342)(172,341)(173,340)(174,339)(175,338)(176,337)(177,336)(178,335)(179,334)(180,333)(181,332)(182,331)(183,330)(184,329)(185,328)(186,327)(187,326)(188,325)(189,324)(190,323)(191,322)(192,321)(193,320)(194,319)(195,318)(196,317)(197,316)(198,315)(199,314)(200,313)(201,312)(202,311)(203,310)(204,309)(205,308)(206,307)(207,306)(208,305)(209,304)(210,303)(211,302)(212,301)(213,300)(214,299)(215,298)(216,297)(217,296)(218,295)(219,294)(220,293)(221,292)(222,291)(223,290)(224,289)(225,288)(226,287)(227,286)(228,285)(229,284)(230,283)(231,282)(232,281)(233,280)(234,279)(235,278)(236,277)(237,276)(238,275)(239,274)(240,273)(241,272)(242,271)(243,270)(244,269)(245,268)(246,267)(247,266)(248,265)(249,264)(250,263)(251,262)(252,261)(253,260)(254,259)(255,258)(256,257)

Polycyclic presentation:

2,31,449|b^a=b^30,c^a=c^448

Conjugacy classes & character table

6961 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂×D₁₃₉₁₉
maximal normal subgroup of	C₂×D₁₃₉₁₉
maximal quotient of	Dic₁₃₉₁₉, C₂×D₁₃₉₁₉

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(27838, 3);

D13919 #42123770