D₂₆₂₁₅ #72289547

order	52430
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral

D₂₆₂₁₅

Splittings

Fitting, derived, Hall	C₅×C₄₉×C₁₀₇ ⋊ C₂
Hall	C₄₉×C₁₀₇ ⋊ D₅
Hall	C₅×C₁₀₇ ⋊ D₄₉
Hall	C₅×C₄₉ ⋊ D₁₀₇
Sylow	C₁₀₇ ⋊ D₂₄₅
Sylow	C₄₉ ⋊ D₅₃₅
Sylow	C₅ ⋊ D₅₂₄₃

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	C₅×C₄₉×C₁₀₇	C₂
Frattini	C₇	D₃₇₄₅
socle	C₅×C₇×C₁₀₇	D₇
LCS	G ▷ C₅×C₄₉×C₁₀₇
UCS	C₁

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 161
transitive degree	μ_t(G) = 26215

A minimal faithful permutation representation — 2 generators on 161 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);(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)(56,161)(57,160)(58,159)(59,158)(60,157)(61,156)(62,155)(63,154)(64,153)(65,152)(66,151)(67,150)(68,149)(69,148)(70,147)(71,146)(72,145)(73,144)(74,143)(75,142)(76,141)(77,140)(78,139)(79,138)(80,137)(81,136)(82,135)(83,134)(84,133)(85,132)(86,131)(87,130)(88,129)(89,128)(90,127)(91,126)(92,125)(93,124)(94,123)(95,122)(96,121)(97,120)(98,119)(99,118)(100,117)(101,116)(102,115)(103,114)(104,113)(105,112)(106,111)(107,110)(108,109)

Polycyclic presentation:

2,5,7,107,7|c^7=e,b^a=b^4,c^a=c^6e^6,d^a=d^106,e^a=e^6

Conjugacy classes & character table

13109 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, 107, 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.4^106);;
gap> SetConjugate(coll, 5, 1, F.5^6);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D26215 #72289547