D₃₁₁₁₅ #72299990

order	62230
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral

D₃₁₁₁₅

Splittings

Fitting, derived, Hall	C₅×C₄₉×C₁₂₇ ⋊ C₂
Hall	C₄₉×C₁₂₇ ⋊ D₅
Hall	C₅×C₁₂₇ ⋊ D₄₉
Hall	C₅×C₄₉ ⋊ D₁₂₇
Sylow	C₁₂₇ ⋊ D₂₄₅
Sylow	C₄₉ ⋊ D₆₃₅
Sylow	C₅ ⋊ D₆₂₂₃

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	C₅×C₄₉×C₁₂₇	C₂
Frattini	C₇	D₄₄₄₅
socle	C₅×C₇×C₁₂₇	D₇
LCS	G ▷ C₅×C₄₉×C₁₂₇
UCS	C₁

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 181
transitive degree	μ_t(G) = 31115

A minimal faithful permutation representation — 2 generators on 181 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);(2,5)(3,4)(7,54)(8,53)(9,52)(10,51)(11,50)(12,49)(13,48)(14,47)(15,46)(16,45)(17,44)(18,43)(19,42)(20,41)(21,40)(22,39)(23,38)(24,37)(25,36)(26,35)(27,34)(28,33)(29,32)(30,31)(56,181)(57,180)(58,179)(59,178)(60,177)(61,176)(62,175)(63,174)(64,173)(65,172)(66,171)(67,170)(68,169)(69,168)(70,167)(71,166)(72,165)(73,164)(74,163)(75,162)(76,161)(77,160)(78,159)(79,158)(80,157)(81,156)(82,155)(83,154)(84,153)(85,152)(86,151)(87,150)(88,149)(89,148)(90,147)(91,146)(92,145)(93,144)(94,143)(95,142)(96,141)(97,140)(98,139)(99,138)(100,137)(101,136)(102,135)(103,134)(104,133)(105,132)(106,131)(107,130)(108,129)(109,128)(110,127)(111,126)(112,125)(113,124)(114,123)(115,122)(116,121)(117,120)(118,119)

Polycyclic presentation:

2,5,7,127,7|c^7=e,b^a=b^4,c^a=c^6e^6,d^a=d^126,e^a=e^6

Conjugacy classes & character table

15559 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, 5, 7, 127, 7]);;
gap> SetPower(coll, 3, F.5);;
gap> SetConjugate(coll, 2, 1, F.2^4);;
gap> SetConjugate(coll, 3, 1, F.3^6*F.5^6);;
gap> SetConjugate(coll, 4, 1, F.4^126);;
gap> SetConjugate(coll, 5, 1, F.5^6);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D31115 #72299990