D₁₀₄₈₅ #35230451

order	20970
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₁₀₄₈₅
unambiguous semidirect product	C₅⋊D₂₀₉₇
unambiguous semidirect product	C₉⋊D₁₁₆₅
unambiguous semidirect product	C₂₃₃⋊D₄₅

Splittings

Fitting, derived, Hall	C₅×C₉×C₂₃₃ ⋊ C₂
Hall	C₉×C₂₃₃ ⋊ D₅
Hall	C₅×C₂₃₃ ⋊ D₉
Sylow	C₂₃₃ ⋊ D₄₅
Hall	C₅×C₉ ⋊ D₂₃₃
Sylow	C₉ ⋊ D₁₁₆₅
Sylow	C₅ ⋊ D₂₀₉₇

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	C₅×C₉×C₂₃₃	C₂
Frattini	C₃	D₃₄₉₅
socle	C₃×C₅×C₂₃₃	D₃
LCS	G ▷ C₅×C₉×C₂₃₃
UCS	C₁

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 247
transitive degree	μ_t(G) = 10485

A minimal faithful permutation representation — 2 generators on 247 points:

(1,2,3,4,5,6,7,8,9)(10,11,12,13,14)(15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247);(2,9)(3,8)(4,7)(5,6)(11,14)(12,13)(16,247)(17,246)(18,245)(19,244)(20,243)(21,242)(22,241)(23,240)(24,239)(25,238)(26,237)(27,236)(28,235)(29,234)(30,233)(31,232)(32,231)(33,230)(34,229)(35,228)(36,227)(37,226)(38,225)(39,224)(40,223)(41,222)(42,221)(43,220)(44,219)(45,218)(46,217)(47,216)(48,215)(49,214)(50,213)(51,212)(52,211)(53,210)(54,209)(55,208)(56,207)(57,206)(58,205)(59,204)(60,203)(61,202)(62,201)(63,200)(64,199)(65,198)(66,197)(67,196)(68,195)(69,194)(70,193)(71,192)(72,191)(73,190)(74,189)(75,188)(76,187)(77,186)(78,185)(79,184)(80,183)(81,182)(82,181)(83,180)(84,179)(85,178)(86,177)(87,176)(88,175)(89,174)(90,173)(91,172)(92,171)(93,170)(94,169)(95,168)(96,167)(97,166)(98,165)(99,164)(100,163)(101,162)(102,161)(103,160)(104,159)(105,158)(106,157)(107,156)(108,155)(109,154)(110,153)(111,152)(112,151)(113,150)(114,149)(115,148)(116,147)(117,146)(118,145)(119,144)(120,143)(121,142)(122,141)(123,140)(124,139)(125,138)(126,137)(127,136)(128,135)(129,134)(130,133)(131,132)

Polycyclic presentation:

2,3,5,233,3|b^3=e^2,b^a=b^2e,c^a=c^4,d^a=d^232,e^a=e^2

Conjugacy classes & character table

5244 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₃₁₄₅₅
maximal quotient of	D₃₁₄₅₅

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20970, 8);

D10485 #35230451