D₂₇₈₆₀ #72291176

order	55720
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₂₇₈₆₀

Splittings

Fitting	C₄×C₅×C₇×C₁₉₉ ⋊ C₂
Hall	C₅×C₇×C₁₉₉ ⋊ D₄
Hall	C₇×C₁₉₉ ⋊ D₂₀
Hall	C₅×C₁₉₉ ⋊ D₂₈
Sylow	C₁₉₉ ⋊ D₁₄₀
Hall	C₅×C₇ ⋊ D₇₉₆
Sylow	C₇ ⋊ D₃₉₈₀
Sylow	C₅ ⋊ D₅₅₇₂

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₆₉₆₅
commutator, socle, cosocle	C₂×C₅×C₇×C₁₉₉	C₂²
Fitting	C₄×C₅×C₇×C₁₉₉	C₂
hypercenter	C₄	D₆₉₆₅
hypocenter	C₅×C₇×C₁₉₉	D₄
LCS	G ▷ C₂×C₅×C₇×C₁₉₉ ▷ C₅×C₇×C₁₉₉
UCS	C₁ ◁ C₂ ◁ C₄

64 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 215
transitive degree	μ_t(G) = 27860

A minimal faithful permutation representation — 2 generators on 215 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);(2,4)(6,9)(7,8)(11,16)(12,15)(13,14)(18,215)(19,214)(20,213)(21,212)(22,211)(23,210)(24,209)(25,208)(26,207)(27,206)(28,205)(29,204)(30,203)(31,202)(32,201)(33,200)(34,199)(35,198)(36,197)(37,196)(38,195)(39,194)(40,193)(41,192)(42,191)(43,190)(44,189)(45,188)(46,187)(47,186)(48,185)(49,184)(50,183)(51,182)(52,181)(53,180)(54,179)(55,178)(56,177)(57,176)(58,175)(59,174)(60,173)(61,172)(62,171)(63,170)(64,169)(65,168)(66,167)(67,166)(68,165)(69,164)(70,163)(71,162)(72,161)(73,160)(74,159)(75,158)(76,157)(77,156)(78,155)(79,154)(80,153)(81,152)(82,151)(83,150)(84,149)(85,148)(86,147)(87,146)(88,145)(89,144)(90,143)(91,142)(92,141)(93,140)(94,139)(95,138)(96,137)(97,136)(98,135)(99,134)(100,133)(101,132)(102,131)(103,130)(104,129)(105,128)(106,127)(107,126)(108,125)(109,124)(110,123)(111,122)(112,121)(113,120)(114,119)(115,118)(116,117)

Polycyclic presentation:

2^3,5,7,199|b^2=c,b^a=bc,d^a=d^4,e^a=e^6,f^a=f^198

Conjugacy classes & character table

13933 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, 199]);;
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.6^198);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D27860 #72291176