D₁₅₅₅₇ #44527487

order	31114
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₁₅₅₅₇
unambiguous semidirect product	C₄₇⋊D₃₃₁
unambiguous semidirect product	C₃₃₁⋊D₄₇

Splittings

Fitting, socle, derived, Hall	C₄₇×C₃₃₁ ⋊ C₂
Sylow	C₃₃₁ ⋊ D₄₇
Sylow	C₄₇ ⋊ D₃₃₁

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₄₇×C₃₃₁	C₂
LCS	G ▷ C₄₇×C₃₃₁
UCS	C₁

8 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 378
transitive degree	μ_t(G) = 15557

A minimal faithful permutation representation — 2 generators on 378 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);(2,47)(3,46)(4,45)(5,44)(6,43)(7,42)(8,41)(9,40)(10,39)(11,38)(12,37)(13,36)(14,35)(15,34)(16,33)(17,32)(18,31)(19,30)(20,29)(21,28)(22,27)(23,26)(24,25)(49,378)(50,377)(51,376)(52,375)(53,374)(54,373)(55,372)(56,371)(57,370)(58,369)(59,368)(60,367)(61,366)(62,365)(63,364)(64,363)(65,362)(66,361)(67,360)(68,359)(69,358)(70,357)(71,356)(72,355)(73,354)(74,353)(75,352)(76,351)(77,350)(78,349)(79,348)(80,347)(81,346)(82,345)(83,344)(84,343)(85,342)(86,341)(87,340)(88,339)(89,338)(90,337)(91,336)(92,335)(93,334)(94,333)(95,332)(96,331)(97,330)(98,329)(99,328)(100,327)(101,326)(102,325)(103,324)(104,323)(105,322)(106,321)(107,320)(108,319)(109,318)(110,317)(111,316)(112,315)(113,314)(114,313)(115,312)(116,311)(117,310)(118,309)(119,308)(120,307)(121,306)(122,305)(123,304)(124,303)(125,302)(126,301)(127,300)(128,299)(129,298)(130,297)(131,296)(132,295)(133,294)(134,293)(135,292)(136,291)(137,290)(138,289)(139,288)(140,287)(141,286)(142,285)(143,284)(144,283)(145,282)(146,281)(147,280)(148,279)(149,278)(150,277)(151,276)(152,275)(153,274)(154,273)(155,272)(156,271)(157,270)(158,269)(159,268)(160,267)(161,266)(162,265)(163,264)(164,263)(165,262)(166,261)(167,260)(168,259)(169,258)(170,257)(171,256)(172,255)(173,254)(174,253)(175,252)(176,251)(177,250)(178,249)(179,248)(180,247)(181,246)(182,245)(183,244)(184,243)(185,242)(186,241)(187,240)(188,239)(189,238)(190,237)(191,236)(192,235)(193,234)(194,233)(195,232)(196,231)(197,230)(198,229)(199,228)(200,227)(201,226)(202,225)(203,224)(204,223)(205,222)(206,221)(207,220)(208,219)(209,218)(210,217)(211,216)(212,215)(213,214)

Polycyclic presentation:

2,47,331|b^a=b^46,c^a=c^330

Conjugacy classes & character table

7780 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂×D₁₅₅₅₇
maximal normal subgroup of	C₂×D₁₅₅₅₇
maximal quotient of	Dic₁₅₅₅₇, C₂×D₁₅₅₅₇

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(31114, 3);

D15557 #44527487