D₂₇₅₇₂ #72290899

order	55144
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₂₇₅₇₂

Splittings

Fitting	C₄×C₆₁×C₁₁₃ ⋊ C₂
Hall	C₆₁×C₁₁₃ ⋊ D₄
Sylow	C₁₁₃ ⋊ D₂₄₄
Sylow	C₆₁ ⋊ D₄₅₂

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₆₈₉₃
commutator, socle, cosocle	C₂×C₆₁×C₁₁₃	C₂²
Fitting	C₄×C₆₁×C₁₁₃	C₂
hypercenter	C₄	D₆₈₉₃
hypocenter	C₆₁×C₁₁₃	D₄
LCS	G ▷ C₂×C₆₁×C₁₁₃ ▷ C₆₁×C₁₁₃
UCS	C₁ ◁ C₂ ◁ C₄

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 178
transitive degree	μ_t(G) = 27572

A minimal faithful permutation representation — 2 generators on 178 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);(2,4)(6,65)(7,64)(8,63)(9,62)(10,61)(11,60)(12,59)(13,58)(14,57)(15,56)(16,55)(17,54)(18,53)(19,52)(20,51)(21,50)(22,49)(23,48)(24,47)(25,46)(26,45)(27,44)(28,43)(29,42)(30,41)(31,40)(32,39)(33,38)(34,37)(35,36)(67,178)(68,177)(69,176)(70,175)(71,174)(72,173)(73,172)(74,171)(75,170)(76,169)(77,168)(78,167)(79,166)(80,165)(81,164)(82,163)(83,162)(84,161)(85,160)(86,159)(87,158)(88,157)(89,156)(90,155)(91,154)(92,153)(93,152)(94,151)(95,150)(96,149)(97,148)(98,147)(99,146)(100,145)(101,144)(102,143)(103,142)(104,141)(105,140)(106,139)(107,138)(108,137)(109,136)(110,135)(111,134)(112,133)(113,132)(114,131)(115,130)(116,129)(117,128)(118,127)(119,126)(120,125)(121,124)(122,123)

Polycyclic presentation:

2^3,61,113|b^2=c,b^a=bc,d^a=d^60,e^a=e^112

Conjugacy classes & character table

13789 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, 61, 113]);;
gap> SetPower(coll, 2, F.3);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 4, 1, F.4^60);;
gap> SetConjugate(coll, 5, 1, F.5^112);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D27572 #72290899