D₈₉₂₀ #72213782

order	17840
automorphism	Aut(G) order 31683840
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₈₉₂₀

Splittings

Fitting	C₅×C₈×C₂₂₃ ⋊ C₂
Hall	C₅×C₂₂₃ ⋊ D₈
Sylow	C₂₂₃ ⋊ D₄₀
Sylow	C₅ ⋊ D₁₇₈₄

Subgroups

	subgroup	G/H
center	C₂	D₄₄₆₀
commutator, cosocle	C₄×C₅×C₂₂₃	C₂²
Frattini	C₄	C₂×D₁₁₁₅
Fitting	C₅×C₈×C₂₂₃	C₂
socle	C₂×C₅×C₂₂₃	D₄
hypercenter	C₈	D₁₁₁₅
hypocenter	C₅×C₂₂₃	D₈
LCS	G ▷ C₄×C₅×C₂₂₃ ▷ C₂×C₅×C₂₂₃ ▷ C₅×C₂₂₃
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

44 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 236
transitive degree	μ_t(G) = 8920

A minimal faithful permutation representation — 2 generators on 236 points:

(1,2,3,4,5,6,7,8)(9,10,11,12,13)(14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236);(2,8)(3,7)(4,6)(10,13)(11,12)(15,236)(16,235)(17,234)(18,233)(19,232)(20,231)(21,230)(22,229)(23,228)(24,227)(25,226)(26,225)(27,224)(28,223)(29,222)(30,221)(31,220)(32,219)(33,218)(34,217)(35,216)(36,215)(37,214)(38,213)(39,212)(40,211)(41,210)(42,209)(43,208)(44,207)(45,206)(46,205)(47,204)(48,203)(49,202)(50,201)(51,200)(52,199)(53,198)(54,197)(55,196)(56,195)(57,194)(58,193)(59,192)(60,191)(61,190)(62,189)(63,188)(64,187)(65,186)(66,185)(67,184)(68,183)(69,182)(70,181)(71,180)(72,179)(73,178)(74,177)(75,176)(76,175)(77,174)(78,173)(79,172)(80,171)(81,170)(82,169)(83,168)(84,167)(85,166)(86,165)(87,164)(88,163)(89,162)(90,161)(91,160)(92,159)(93,158)(94,157)(95,156)(96,155)(97,154)(98,153)(99,152)(100,151)(101,150)(102,149)(103,148)(104,147)(105,146)(106,145)(107,144)(108,143)(109,142)(110,141)(111,140)(112,139)(113,138)(114,137)(115,136)(116,135)(117,134)(118,133)(119,132)(120,131)(121,130)(122,129)(123,128)(124,127)(125,126)

Polycyclic presentation:

2^4,5,223|b^2=cd,c^2=d,b^a=bc,c^a=cd,e^a=e^4,f^a=f^222

Conjugacy classes & character table

4463 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₁₇₈₄₀, D₂₆₇₆₀
maximal normal subgroup of	D₁₇₈₄₀
maximal quotient of	Dic₈₉₂₀, D₁₇₈₄₀, D₂₆₇₆₀

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, 2, 5, 223]);;
gap> SetPower(coll, 2, F.3*F.4);;
gap> SetPower(coll, 3, F.4);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 5, 1, F.5^4);;
gap> SetConjugate(coll, 6, 1, F.6^222);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D8920 #72213782