D₁₃₄₂₇ #41001672

order	26854
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₁₃₄₂₇
unambiguous semidirect product	C₂₉⋊D₄₆₃
unambiguous semidirect product	C₄₆₃⋊D₂₉

Splittings

Fitting, socle, derived, Hall	C₂₉×C₄₆₃ ⋊ C₂
Sylow	C₄₆₃ ⋊ D₂₉
Sylow	C₂₉ ⋊ D₄₆₃

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₂₉×C₄₆₃	C₂
LCS	G ▷ C₂₉×C₄₆₃
UCS	C₁

8 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 492
transitive degree	μ_t(G) = 13427

A minimal faithful permutation representation — 2 generators on 492 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)(30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456,457,458,459,460,461,462,463,464,465,466,467,468,469,470,471,472,473,474,475,476,477,478,479,480,481,482,483,484,485,486,487,488,489,490,491,492);(2,29)(3,28)(4,27)(5,26)(6,25)(7,24)(8,23)(9,22)(10,21)(11,20)(12,19)(13,18)(14,17)(15,16)(31,492)(32,491)(33,490)(34,489)(35,488)(36,487)(37,486)(38,485)(39,484)(40,483)(41,482)(42,481)(43,480)(44,479)(45,478)(46,477)(47,476)(48,475)(49,474)(50,473)(51,472)(52,471)(53,470)(54,469)(55,468)(56,467)(57,466)(58,465)(59,464)(60,463)(61,462)(62,461)(63,460)(64,459)(65,458)(66,457)(67,456)(68,455)(69,454)(70,453)(71,452)(72,451)(73,450)(74,449)(75,448)(76,447)(77,446)(78,445)(79,444)(80,443)(81,442)(82,441)(83,440)(84,439)(85,438)(86,437)(87,436)(88,435)(89,434)(90,433)(91,432)(92,431)(93,430)(94,429)(95,428)(96,427)(97,426)(98,425)(99,424)(100,423)(101,422)(102,421)(103,420)(104,419)(105,418)(106,417)(107,416)(108,415)(109,414)(110,413)(111,412)(112,411)(113,410)(114,409)(115,408)(116,407)(117,406)(118,405)(119,404)(120,403)(121,402)(122,401)(123,400)(124,399)(125,398)(126,397)(127,396)(128,395)(129,394)(130,393)(131,392)(132,391)(133,390)(134,389)(135,388)(136,387)(137,386)(138,385)(139,384)(140,383)(141,382)(142,381)(143,380)(144,379)(145,378)(146,377)(147,376)(148,375)(149,374)(150,373)(151,372)(152,371)(153,370)(154,369)(155,368)(156,367)(157,366)(158,365)(159,364)(160,363)(161,362)(162,361)(163,360)(164,359)(165,358)(166,357)(167,356)(168,355)(169,354)(170,353)(171,352)(172,351)(173,350)(174,349)(175,348)(176,347)(177,346)(178,345)(179,344)(180,343)(181,342)(182,341)(183,340)(184,339)(185,338)(186,337)(187,336)(188,335)(189,334)(190,333)(191,332)(192,331)(193,330)(194,329)(195,328)(196,327)(197,326)(198,325)(199,324)(200,323)(201,322)(202,321)(203,320)(204,319)(205,318)(206,317)(207,316)(208,315)(209,314)(210,313)(211,312)(212,311)(213,310)(214,309)(215,308)(216,307)(217,306)(218,305)(219,304)(220,303)(221,302)(222,301)(223,300)(224,299)(225,298)(226,297)(227,296)(228,295)(229,294)(230,293)(231,292)(232,291)(233,290)(234,289)(235,288)(236,287)(237,286)(238,285)(239,284)(240,283)(241,282)(242,281)(243,280)(244,279)(245,278)(246,277)(247,276)(248,275)(249,274)(250,273)(251,272)(252,271)(253,270)(254,269)(255,268)(256,267)(257,266)(258,265)(259,264)(260,263)(261,262)

Polycyclic presentation:

2,29,463|b^a=b^28,c^a=c^462

Conjugacy classes & character table

6715 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂×D₁₃₄₂₇
maximal normal subgroup of	C₂×D₁₃₄₂₇
maximal quotient of	Dic₁₃₄₂₇, C₂×D₁₃₄₂₇

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(26854, 3);

D13427 #41001672