Dic₁₄₄₉₅ #72291222

order	57980
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₄₄₉₅

Splittings

derived, Hall	C₅×C₁₃×C₂₂₃ ⋊ C₄
Hall	C₁₃×C₂₂₃ ⋊ Dic₅
Hall	C₅×C₂₂₃ ⋊ Dic₁₃
Sylow	C₂₂₃ ⋊ Dic₆₅
Hall	C₅×C₁₃ ⋊ Dic₂₂₃
Sylow	C₁₃ ⋊ Dic₁₁₁₅
Sylow	C₅ ⋊ Dic₂₈₉₉

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₁₄₄₉₅
commutator, hypocenter	C₅×C₁₃×C₂₂₃	C₄
Fitting, socle, prime radical, cosocle	C₂×C₅×C₁₃×C₂₂₃	C₂
LCS	G ▷ C₅×C₁₃×C₂₂₃
UCS	C₁ ◁ C₂

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 245
transitive degree	μ_t(G) = 57980

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

Polycyclic presentation:

2^2,5,13,223|a^2=b,c^a=c^4,d^a=d^12,e^a=e^222

Conjugacy classes & character table

14498 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");;
gap> coll := SingleCollector(F, [2, 2, 5, 13, 223]);;
gap> SetPower(coll, 1, F.2);;
gap> SetConjugate(coll, 3, 1, F.3^4);;
gap> SetConjugate(coll, 4, 1, F.4^12);;
gap> SetConjugate(coll, 5, 1, F.5^222);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic14495 #72291222