Dic₁₅₄₂₀ #72293507

order	61680
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₅₄₂₀

Splittings

Hall	C₃×C₅×C₂₅₇ ⋊ Q₁₆
Hall	C₅×C₂₅₇ ⋊ Dic₁₂
Hall	C₃×C₂₅₇ ⋊ Dic₂₀
Sylow	C₂₅₇ ⋊ Dic₆₀
Hall	C₃×C₅ ⋊ Dic₁₀₂₈
Sylow	C₅ ⋊ Dic₃₀₈₄
Sylow	C₃ ⋊ Dic₅₁₄₀

Subgroups

	subgroup	G/H
center	C₂	D₁₅₄₂₀
commutator, cosocle	C₃×C₄×C₅×C₂₅₇	C₂²
Frattini	C₄	C₂×D₃₈₅₅
Fitting	C₃×C₅×C₈×C₂₅₇	C₂
socle, prime radical	C₂×C₃×C₅×C₂₅₇	D₄
hypercenter	C₈	D₃₈₅₅
hypocenter	C₃×C₅×C₂₅₇	Q₁₆
LCS	G ▷ C₃×C₄×C₅×C₂₅₇ ▷ C₂×C₃×C₅×C₂₅₇ ▷ C₃×C₅×C₂₅₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

72 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 281
transitive degree	μ_t(G) = 61680

A minimal faithful permutation representation — 2 generators on 281 points:

(1,2,3,4,5,6,7,8)(9,16,15,14,13,12,11,10)(17,18,19)(20,21,22,23,24)(25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281);(1,9,5,13)(2,10,6,14)(3,11,7,15)(4,12,8,16)(18,19)(21,24)(22,23)(26,281)(27,280)(28,279)(29,278)(30,277)(31,276)(32,275)(33,274)(34,273)(35,272)(36,271)(37,270)(38,269)(39,268)(40,267)(41,266)(42,265)(43,264)(44,263)(45,262)(46,261)(47,260)(48,259)(49,258)(50,257)(51,256)(52,255)(53,254)(54,253)(55,252)(56,251)(57,250)(58,249)(59,248)(60,247)(61,246)(62,245)(63,244)(64,243)(65,242)(66,241)(67,240)(68,239)(69,238)(70,237)(71,236)(72,235)(73,234)(74,233)(75,232)(76,231)(77,230)(78,229)(79,228)(80,227)(81,226)(82,225)(83,224)(84,223)(85,222)(86,221)(87,220)(88,219)(89,218)(90,217)(91,216)(92,215)(93,214)(94,213)(95,212)(96,211)(97,210)(98,209)(99,208)(100,207)(101,206)(102,205)(103,204)(104,203)(105,202)(106,201)(107,200)(108,199)(109,198)(110,197)(111,196)(112,195)(113,194)(114,193)(115,192)(116,191)(117,190)(118,189)(119,188)(120,187)(121,186)(122,185)(123,184)(124,183)(125,182)(126,181)(127,180)(128,179)(129,178)(130,177)(131,176)(132,175)(133,174)(134,173)(135,172)(136,171)(137,170)(138,169)(139,168)(140,167)(141,166)(142,165)(143,164)(144,163)(145,162)(146,161)(147,160)(148,159)(149,158)(150,157)(151,156)(152,155)(153,154)

Polycyclic presentation:

2^4,3,5,257|a^2=d,b^2=cd,c^2=d,b^a=bc,c^a=cd,e^a=e^2,f^a=f^4,g^a=g^256

Conjugacy classes & character table

15423 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", "g");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 3, 5, 257]);;
gap> SetPower(coll, 1, F.4);;
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.5^2);;
gap> SetConjugate(coll, 6, 1, F.6^4);;
gap> SetConjugate(coll, 7, 1, F.7^256);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic15420 #72293507