D₂₆₇₄₀ #72290086

order	53480
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₂₆₇₄₀

Splittings

Fitting	C₄×C₅×C₇×C₁₉₁ ⋊ C₂
Hall	C₅×C₇×C₁₉₁ ⋊ D₄
Hall	C₇×C₁₉₁ ⋊ D₂₀
Hall	C₅×C₁₉₁ ⋊ D₂₈
Sylow	C₁₉₁ ⋊ D₁₄₀
Hall	C₅×C₇ ⋊ D₇₆₄
Sylow	C₇ ⋊ D₃₈₂₀
Sylow	C₅ ⋊ D₅₃₄₈

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₆₆₈₅
commutator, socle, cosocle	C₂×C₅×C₇×C₁₉₁	C₂²
Fitting	C₄×C₅×C₇×C₁₉₁	C₂
hypercenter	C₄	D₆₆₈₅
hypocenter	C₅×C₇×C₁₉₁	D₄
LCS	G ▷ C₂×C₅×C₇×C₁₉₁ ▷ C₅×C₇×C₁₉₁
UCS	C₁ ◁ C₂ ◁ C₄

64 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 207
transitive degree	μ_t(G) = 26740

A minimal faithful permutation representation — 2 generators on 207 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);(2,4)(6,9)(7,8)(11,16)(12,15)(13,14)(18,207)(19,206)(20,205)(21,204)(22,203)(23,202)(24,201)(25,200)(26,199)(27,198)(28,197)(29,196)(30,195)(31,194)(32,193)(33,192)(34,191)(35,190)(36,189)(37,188)(38,187)(39,186)(40,185)(41,184)(42,183)(43,182)(44,181)(45,180)(46,179)(47,178)(48,177)(49,176)(50,175)(51,174)(52,173)(53,172)(54,171)(55,170)(56,169)(57,168)(58,167)(59,166)(60,165)(61,164)(62,163)(63,162)(64,161)(65,160)(66,159)(67,158)(68,157)(69,156)(70,155)(71,154)(72,153)(73,152)(74,151)(75,150)(76,149)(77,148)(78,147)(79,146)(80,145)(81,144)(82,143)(83,142)(84,141)(85,140)(86,139)(87,138)(88,137)(89,136)(90,135)(91,134)(92,133)(93,132)(94,131)(95,130)(96,129)(97,128)(98,127)(99,126)(100,125)(101,124)(102,123)(103,122)(104,121)(105,120)(106,119)(107,118)(108,117)(109,116)(110,115)(111,114)(112,113)

Polycyclic presentation:

2^3,5,7,191|b^2=c,b^a=bc,d^a=d^4,e^a=e^6,f^a=f^190

Conjugacy classes & character table

13373 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, 2, 2, 5, 7, 191]);;
gap> SetPower(coll, 2, F.3);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 4, 1, F.4^4);;
gap> SetConjugate(coll, 5, 1, F.5^6);;
gap> SetConjugate(coll, 6, 1, F.6^190);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D26740 #72290086