D₂₉₂₅₃ #72293199

order	58506
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral

D₂₉₂₅₃

Splittings

Fitting, derived, Hall	C₃×C₄₉×C₁₉₉ ⋊ C₂
Hall	C₄₉×C₁₉₉ ⋊ D₃
Hall	C₃×C₁₉₉ ⋊ D₄₉
Sylow	C₁₉₉ ⋊ D₁₄₇
Hall	C₃×C₄₉ ⋊ D₁₉₉
Sylow	C₄₉ ⋊ D₅₉₇
Sylow	C₃ ⋊ D₉₇₅₁

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	C₃×C₄₉×C₁₉₉	C₂
Frattini	C₇	D₄₁₇₉
socle	C₃×C₇×C₁₉₉	D₇
LCS	G ▷ C₃×C₄₉×C₁₉₉
UCS	C₁

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 251
transitive degree	μ_t(G) = 29253

A minimal faithful permutation representation — 2 generators on 251 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);(2,3)(5,52)(6,51)(7,50)(8,49)(9,48)(10,47)(11,46)(12,45)(13,44)(14,43)(15,42)(16,41)(17,40)(18,39)(19,38)(20,37)(21,36)(22,35)(23,34)(24,33)(25,32)(26,31)(27,30)(28,29)(54,251)(55,250)(56,249)(57,248)(58,247)(59,246)(60,245)(61,244)(62,243)(63,242)(64,241)(65,240)(66,239)(67,238)(68,237)(69,236)(70,235)(71,234)(72,233)(73,232)(74,231)(75,230)(76,229)(77,228)(78,227)(79,226)(80,225)(81,224)(82,223)(83,222)(84,221)(85,220)(86,219)(87,218)(88,217)(89,216)(90,215)(91,214)(92,213)(93,212)(94,211)(95,210)(96,209)(97,208)(98,207)(99,206)(100,205)(101,204)(102,203)(103,202)(104,201)(105,200)(106,199)(107,198)(108,197)(109,196)(110,195)(111,194)(112,193)(113,192)(114,191)(115,190)(116,189)(117,188)(118,187)(119,186)(120,185)(121,184)(122,183)(123,182)(124,181)(125,180)(126,179)(127,178)(128,177)(129,176)(130,175)(131,174)(132,173)(133,172)(134,171)(135,170)(136,169)(137,168)(138,167)(139,166)(140,165)(141,164)(142,163)(143,162)(144,161)(145,160)(146,159)(147,158)(148,157)(149,156)(150,155)(151,154)(152,153)

Polycyclic presentation:

2,3,7,199,7|c^7=e,b^a=b^2,c^a=c^6e^6,d^a=d^198,e^a=e^6

Conjugacy classes & character table

14628 conjugacy classes. The character table is too large to display.

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e");;
gap> coll := SingleCollector(F, [2, 3, 7, 199, 7]);;
gap> SetPower(coll, 3, F.5);;
gap> SetConjugate(coll, 2, 1, F.2^2);;
gap> SetConjugate(coll, 3, 1, F.3^6*F.5^6);;
gap> SetConjugate(coll, 4, 1, F.4^198);;
gap> SetConjugate(coll, 5, 1, F.5^6);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D29253 #72293199