D₃₀₀₇₆ #72295401

order	60152
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₃₀₀₇₆

Splittings

Fitting	C₄×C₇₃×C₁₀₃ ⋊ C₂
Hall	C₇₃×C₁₀₃ ⋊ D₄
Sylow	C₁₀₃ ⋊ D₂₉₂
Sylow	C₇₃ ⋊ D₄₁₂

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₇₅₁₉
commutator, socle, cosocle	C₂×C₇₃×C₁₀₃	C₂²
Fitting	C₄×C₇₃×C₁₀₃	C₂
hypercenter	C₄	D₇₅₁₉
hypocenter	C₇₃×C₁₀₃	D₄
LCS	G ▷ C₂×C₇₃×C₁₀₃ ▷ C₇₃×C₁₀₃
UCS	C₁ ◁ C₂ ◁ C₄

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 180
transitive degree	μ_t(G) = 30076

A minimal faithful permutation representation — 2 generators on 180 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);(2,4)(6,77)(7,76)(8,75)(9,74)(10,73)(11,72)(12,71)(13,70)(14,69)(15,68)(16,67)(17,66)(18,65)(19,64)(20,63)(21,62)(22,61)(23,60)(24,59)(25,58)(26,57)(27,56)(28,55)(29,54)(30,53)(31,52)(32,51)(33,50)(34,49)(35,48)(36,47)(37,46)(38,45)(39,44)(40,43)(41,42)(79,180)(80,179)(81,178)(82,177)(83,176)(84,175)(85,174)(86,173)(87,172)(88,171)(89,170)(90,169)(91,168)(92,167)(93,166)(94,165)(95,164)(96,163)(97,162)(98,161)(99,160)(100,159)(101,158)(102,157)(103,156)(104,155)(105,154)(106,153)(107,152)(108,151)(109,150)(110,149)(111,148)(112,147)(113,146)(114,145)(115,144)(116,143)(117,142)(118,141)(119,140)(120,139)(121,138)(122,137)(123,136)(124,135)(125,134)(126,133)(127,132)(128,131)(129,130)

Polycyclic presentation:

2^3,73,103|b^2=c,b^a=bc,d^a=d^72,e^a=e^102

Conjugacy classes & character table

15041 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, 2, 2, 73, 103]);;
gap> SetPower(coll, 2, F.3);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 4, 1, F.4^72);;
gap> SetConjugate(coll, 5, 1, F.5^102);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D30076 #72295401