D₂₆₃₁₃ #72289672

order	52626
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral

D₂₆₃₁₃

Splittings

Fitting, derived, Hall	C₃×C₄₉×C₁₇₉ ⋊ C₂
Hall	C₄₉×C₁₇₉ ⋊ D₃
Hall	C₃×C₁₇₉ ⋊ D₄₉
Sylow	C₁₇₉ ⋊ D₁₄₇
Hall	C₃×C₄₉ ⋊ D₁₇₉
Sylow	C₄₉ ⋊ D₅₃₇
Sylow	C₃ ⋊ D₈₇₇₁

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	C₃×C₄₉×C₁₇₉	C₂
Frattini	C₇	D₃₇₅₉
socle	C₃×C₇×C₁₇₉	D₇
LCS	G ▷ C₃×C₄₉×C₁₇₉
UCS	C₁

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 231
transitive degree	μ_t(G) = 26313

A minimal faithful permutation representation — 2 generators on 231 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);(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)(54,231)(55,230)(56,229)(57,228)(58,227)(59,226)(60,225)(61,224)(62,223)(63,222)(64,221)(65,220)(66,219)(67,218)(68,217)(69,216)(70,215)(71,214)(72,213)(73,212)(74,211)(75,210)(76,209)(77,208)(78,207)(79,206)(80,205)(81,204)(82,203)(83,202)(84,201)(85,200)(86,199)(87,198)(88,197)(89,196)(90,195)(91,194)(92,193)(93,192)(94,191)(95,190)(96,189)(97,188)(98,187)(99,186)(100,185)(101,184)(102,183)(103,182)(104,181)(105,180)(106,179)(107,178)(108,177)(109,176)(110,175)(111,174)(112,173)(113,172)(114,171)(115,170)(116,169)(117,168)(118,167)(119,166)(120,165)(121,164)(122,163)(123,162)(124,161)(125,160)(126,159)(127,158)(128,157)(129,156)(130,155)(131,154)(132,153)(133,152)(134,151)(135,150)(136,149)(137,148)(138,147)(139,146)(140,145)(141,144)(142,143)

Polycyclic presentation:

2,3,7,179,7|c^7=e,b^a=b^2,c^a=c^6e^6,d^a=d^178,e^a=e^6

Conjugacy classes & character table

13158 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, 179, 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.4^178);;
gap> SetConjugate(coll, 5, 1, F.5^6);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D26313 #72289672