D₁₅₂₈₀ #72271995

order	30560
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₁₅₂₈₀

Splittings

Fitting	C₅×C₁₆×C₁₉₁ ⋊ C₂
Hall	C₅×C₁₉₁ ⋊ D₁₆
Sylow	C₁₉₁ ⋊ D₈₀
Sylow	C₅ ⋊ D₃₀₅₆

Subgroups

	subgroup	G/H
center	C₂	D₇₆₄₀
commutator, cosocle	C₅×C₈×C₁₉₁	C₂²
Frattini	C₈	C₂×D₉₅₅
Fitting	C₅×C₁₆×C₁₉₁	C₂
socle	C₂×C₅×C₁₉₁	D₈
hypercenter	C₁₆	D₉₅₅
hypocenter	C₅×C₁₉₁	D₁₆
LCS	G ▷ C₅×C₈×C₁₉₁ ▷ C₄×C₅×C₁₉₁ ▷ C₂×C₅×C₁₉₁ ▷ C₅×C₁₉₁
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆

56 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 212
transitive degree	μ_t(G) = 15280

A minimal faithful permutation representation — 2 generators on 212 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);(2,16)(3,15)(4,14)(5,13)(6,12)(7,11)(8,10)(18,21)(19,20)(23,212)(24,211)(25,210)(26,209)(27,208)(28,207)(29,206)(30,205)(31,204)(32,203)(33,202)(34,201)(35,200)(36,199)(37,198)(38,197)(39,196)(40,195)(41,194)(42,193)(43,192)(44,191)(45,190)(46,189)(47,188)(48,187)(49,186)(50,185)(51,184)(52,183)(53,182)(54,181)(55,180)(56,179)(57,178)(58,177)(59,176)(60,175)(61,174)(62,173)(63,172)(64,171)(65,170)(66,169)(67,168)(68,167)(69,166)(70,165)(71,164)(72,163)(73,162)(74,161)(75,160)(76,159)(77,158)(78,157)(79,156)(80,155)(81,154)(82,153)(83,152)(84,151)(85,150)(86,149)(87,148)(88,147)(89,146)(90,145)(91,144)(92,143)(93,142)(94,141)(95,140)(96,139)(97,138)(98,137)(99,136)(100,135)(101,134)(102,133)(103,132)(104,131)(105,130)(106,129)(107,128)(108,127)(109,126)(110,125)(111,124)(112,123)(113,122)(114,121)(115,120)(116,119)(117,118)

Polycyclic presentation:

2^5,5,191|b^2=cd,c^2=de,d^2=e,b^a=bc,c^a=cd,d^a=de,f^a=f^4,g^a=g^190

Conjugacy classes & character table

7643 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₃₀₅₆₀
maximal normal subgroup of	D₃₀₅₆₀
maximal quotient of	Dic₁₅₂₈₀, D₃₀₅₆₀

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

D15280 #72271995