D₁₅₃₃₇ #44501555

order	30674
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₁₅₃₃₇
unambiguous semidirect product	C₄₉⋊D₃₁₃
unambiguous semidirect product	C₃₁₃⋊D₄₉

Splittings

Fitting, derived, Hall	C₄₉×C₃₁₃ ⋊ C₂
Sylow	C₃₁₃ ⋊ D₄₉
Sylow	C₄₉ ⋊ D₃₁₃

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	C₄₉×C₃₁₃	C₂
Frattini	C₇	D₂₁₉₁
socle	C₇×C₃₁₃	D₇
LCS	G ▷ C₄₉×C₃₁₃
UCS	C₁

12 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 362
transitive degree	μ_t(G) = 15337

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

Polycyclic presentation:

2,7,313,7|b^7=d^6,b^a=b^6d,c^a=c^312,d^a=d^6

Conjugacy classes & character table

7670 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂×D₁₅₃₃₇
maximal normal subgroup of	C₂×D₁₅₃₃₇
maximal quotient of	Dic₁₅₃₃₇, C₂×D₁₅₃₃₇

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(30674, 4);

D15337 #44501555