D₁₇₄₅₁ #46853174

order	34902
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₁₇₄₅₁
unambiguous semidirect product	C₇⋊D₂₄₉₃
unambiguous semidirect product	C₉⋊D₁₉₃₉
unambiguous semidirect product	C₂₇₇⋊D₆₃

Splittings

Fitting, derived, Hall	C₇×C₉×C₂₇₇ ⋊ C₂
Hall	C₉×C₂₇₇ ⋊ D₇
Hall	C₇×C₂₇₇ ⋊ D₉
Sylow	C₂₇₇ ⋊ D₆₃
Hall	C₇×C₉ ⋊ D₂₇₇
Sylow	C₉ ⋊ D₁₉₃₉
Sylow	C₇ ⋊ D₂₄₉₃

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	C₇×C₉×C₂₇₇	C₂
Frattini	C₃	D₅₈₁₇
socle	C₃×C₇×C₂₇₇	D₃
LCS	G ▷ C₇×C₉×C₂₇₇
UCS	C₁

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 293
transitive degree	μ_t(G) = 17451

A minimal faithful permutation representation — 2 generators on 293 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);(2,9)(3,8)(4,7)(5,6)(11,16)(12,15)(13,14)(18,293)(19,292)(20,291)(21,290)(22,289)(23,288)(24,287)(25,286)(26,285)(27,284)(28,283)(29,282)(30,281)(31,280)(32,279)(33,278)(34,277)(35,276)(36,275)(37,274)(38,273)(39,272)(40,271)(41,270)(42,269)(43,268)(44,267)(45,266)(46,265)(47,264)(48,263)(49,262)(50,261)(51,260)(52,259)(53,258)(54,257)(55,256)(56,255)(57,254)(58,253)(59,252)(60,251)(61,250)(62,249)(63,248)(64,247)(65,246)(66,245)(67,244)(68,243)(69,242)(70,241)(71,240)(72,239)(73,238)(74,237)(75,236)(76,235)(77,234)(78,233)(79,232)(80,231)(81,230)(82,229)(83,228)(84,227)(85,226)(86,225)(87,224)(88,223)(89,222)(90,221)(91,220)(92,219)(93,218)(94,217)(95,216)(96,215)(97,214)(98,213)(99,212)(100,211)(101,210)(102,209)(103,208)(104,207)(105,206)(106,205)(107,204)(108,203)(109,202)(110,201)(111,200)(112,199)(113,198)(114,197)(115,196)(116,195)(117,194)(118,193)(119,192)(120,191)(121,190)(122,189)(123,188)(124,187)(125,186)(126,185)(127,184)(128,183)(129,182)(130,181)(131,180)(132,179)(133,178)(134,177)(135,176)(136,175)(137,174)(138,173)(139,172)(140,171)(141,170)(142,169)(143,168)(144,167)(145,166)(146,165)(147,164)(148,163)(149,162)(150,161)(151,160)(152,159)(153,158)(154,157)(155,156)

Polycyclic presentation:

2,3,7,277,3|b^3=e^2,b^a=b^2e,c^a=c^6,d^a=d^276,e^a=e^2

Conjugacy classes & character table

8727 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(34902, 12);

D17451 #46853174