D₂₅₇₈₅ #72288923

order	51570
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral

D₂₅₇₈₅

Splittings

Fitting, derived, Hall	C₅×C₂₇×C₁₉₁ ⋊ C₂
Hall	C₂₇×C₁₉₁ ⋊ D₅
Hall	C₅×C₁₉₁ ⋊ D₂₇
Sylow	C₁₉₁ ⋊ D₁₃₅
Hall	C₅×C₂₇ ⋊ D₁₉₁
Sylow	C₂₇ ⋊ D₉₅₅
Sylow	C₅ ⋊ D₅₁₅₇

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	C₅×C₂₇×C₁₉₁	C₂
Frattini	C₉	D₂₈₆₅
socle	C₃×C₅×C₁₉₁	D₉
LCS	G ▷ C₅×C₂₇×C₁₉₁
UCS	C₁

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 223
transitive degree	μ_t(G) = 25785

A minimal faithful permutation representation — 2 generators on 223 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);(2,27)(3,26)(4,25)(5,24)(6,23)(7,22)(8,21)(9,20)(10,19)(11,18)(12,17)(13,16)(14,15)(29,32)(30,31)(34,223)(35,222)(36,221)(37,220)(38,219)(39,218)(40,217)(41,216)(42,215)(43,214)(44,213)(45,212)(46,211)(47,210)(48,209)(49,208)(50,207)(51,206)(52,205)(53,204)(54,203)(55,202)(56,201)(57,200)(58,199)(59,198)(60,197)(61,196)(62,195)(63,194)(64,193)(65,192)(66,191)(67,190)(68,189)(69,188)(70,187)(71,186)(72,185)(73,184)(74,183)(75,182)(76,181)(77,180)(78,179)(79,178)(80,177)(81,176)(82,175)(83,174)(84,173)(85,172)(86,171)(87,170)(88,169)(89,168)(90,167)(91,166)(92,165)(93,164)(94,163)(95,162)(96,161)(97,160)(98,159)(99,158)(100,157)(101,156)(102,155)(103,154)(104,153)(105,152)(106,151)(107,150)(108,149)(109,148)(110,147)(111,146)(112,145)(113,144)(114,143)(115,142)(116,141)(117,140)(118,139)(119,138)(120,137)(121,136)(122,135)(123,134)(124,133)(125,132)(126,131)(127,130)(128,129)

Polycyclic presentation:

2,3,5,191,3^2|b^3=e,e^3=f,b^a=b^2e^2f^2,c^a=c^4,d^a=d^190,e^a=e^2f^2,f^a=f^2

Conjugacy classes & character table

12894 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", "f");;
gap> coll := SingleCollector(F, [2, 3, 5, 191, 3, 3]);;
gap> SetPower(coll, 2, F.5);;
gap> SetPower(coll, 5, F.6);;
gap> SetConjugate(coll, 2, 1, F.2^2*F.5^2*F.6^2);;
gap> SetConjugate(coll, 3, 1, F.3^4);;
gap> SetConjugate(coll, 4, 1, F.4^190);;
gap> SetConjugate(coll, 5, 1, F.5^2*F.6^2);;
gap> SetConjugate(coll, 6, 1, F.6^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D25785 #72288923