Dic₁₄₉₀₆ #72292247

order	59624
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₄₉₀₆

Splittings

Hall	C₂₉×C₂₅₇ ⋊ Q₈
Sylow	C₂₅₇ ⋊ Dic₅₈
Sylow	C₂₉ ⋊ Dic₅₁₄

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₇₄₅₃
commutator, socle, prime radical, cosocle	C₂×C₂₉×C₂₅₇	C₂²
Fitting	C₄×C₂₉×C₂₅₇	C₂
hypercenter	C₄	D₇₄₅₃
hypocenter	C₂₉×C₂₅₇	Q₈
LCS	G ▷ C₂×C₂₉×C₂₅₇ ▷ C₂₉×C₂₅₇
UCS	C₁ ◁ C₂ ◁ C₄

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 294
transitive degree	μ_t(G) = 59624

A minimal faithful permutation representation — 2 generators on 294 points:

(1,2,3,4)(5,8,7,6)(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)(38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294);(1,5,3,7)(2,6,4,8)(10,37)(11,36)(12,35)(13,34)(14,33)(15,32)(16,31)(17,30)(18,29)(19,28)(20,27)(21,26)(22,25)(23,24)(39,294)(40,293)(41,292)(42,291)(43,290)(44,289)(45,288)(46,287)(47,286)(48,285)(49,284)(50,283)(51,282)(52,281)(53,280)(54,279)(55,278)(56,277)(57,276)(58,275)(59,274)(60,273)(61,272)(62,271)(63,270)(64,269)(65,268)(66,267)(67,266)(68,265)(69,264)(70,263)(71,262)(72,261)(73,260)(74,259)(75,258)(76,257)(77,256)(78,255)(79,254)(80,253)(81,252)(82,251)(83,250)(84,249)(85,248)(86,247)(87,246)(88,245)(89,244)(90,243)(91,242)(92,241)(93,240)(94,239)(95,238)(96,237)(97,236)(98,235)(99,234)(100,233)(101,232)(102,231)(103,230)(104,229)(105,228)(106,227)(107,226)(108,225)(109,224)(110,223)(111,222)(112,221)(113,220)(114,219)(115,218)(116,217)(117,216)(118,215)(119,214)(120,213)(121,212)(122,211)(123,210)(124,209)(125,208)(126,207)(127,206)(128,205)(129,204)(130,203)(131,202)(132,201)(133,200)(134,199)(135,198)(136,197)(137,196)(138,195)(139,194)(140,193)(141,192)(142,191)(143,190)(144,189)(145,188)(146,187)(147,186)(148,185)(149,184)(150,183)(151,182)(152,181)(153,180)(154,179)(155,178)(156,177)(157,176)(158,175)(159,174)(160,173)(161,172)(162,171)(163,170)(164,169)(165,168)(166,167)

Polycyclic presentation:

2^3,29,257|a^2=c,b^2=c,b^a=bc,d^a=d^28,e^a=e^256

Conjugacy classes & character table

14909 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, 2, 29, 257]);;
gap> SetPower(coll, 1, F.3);;
gap> SetPower(coll, 2, F.3);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 4, 1, F.4^28);;
gap> SetConjugate(coll, 5, 1, F.5^256);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic14906 #72292247