Dic₁₃₅₅₀ #72289655

order	54200
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₃₅₅₀

Splittings

Hall	C₂₅×C₂₇₁ ⋊ Q₈
Sylow	C₂₇₁ ⋊ Dic₅₀
Sylow	C₂₅ ⋊ Dic₅₄₂

Subgroups

	subgroup	G/H
center	C₂	C₂×D₆₇₇₅
commutator, cosocle	C₂×C₂₅×C₂₇₁	C₂²
Frattini	C₂×C₅	C₂×D₁₃₅₅
Fitting	C₄×C₂₅×C₂₇₁	C₂
socle, prime radical	C₂×C₅×C₂₇₁	C₂×D₅
hypercenter	C₄	D₆₇₇₅
hypocenter	C₂₅×C₂₇₁	Q₈
LCS	G ▷ C₂×C₂₅×C₂₇₁ ▷ C₂₅×C₂₇₁
UCS	C₁ ◁ C₂ ◁ C₄

36 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 304
transitive degree	μ_t(G) = 54200

A minimal faithful permutation representation — 2 generators on 304 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)(34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304);(1,5,3,7)(2,6,4,8)(10,33)(11,32)(12,31)(13,30)(14,29)(15,28)(16,27)(17,26)(18,25)(19,24)(20,23)(21,22)(35,304)(36,303)(37,302)(38,301)(39,300)(40,299)(41,298)(42,297)(43,296)(44,295)(45,294)(46,293)(47,292)(48,291)(49,290)(50,289)(51,288)(52,287)(53,286)(54,285)(55,284)(56,283)(57,282)(58,281)(59,280)(60,279)(61,278)(62,277)(63,276)(64,275)(65,274)(66,273)(67,272)(68,271)(69,270)(70,269)(71,268)(72,267)(73,266)(74,265)(75,264)(76,263)(77,262)(78,261)(79,260)(80,259)(81,258)(82,257)(83,256)(84,255)(85,254)(86,253)(87,252)(88,251)(89,250)(90,249)(91,248)(92,247)(93,246)(94,245)(95,244)(96,243)(97,242)(98,241)(99,240)(100,239)(101,238)(102,237)(103,236)(104,235)(105,234)(106,233)(107,232)(108,231)(109,230)(110,229)(111,228)(112,227)(113,226)(114,225)(115,224)(116,223)(117,222)(118,221)(119,220)(120,219)(121,218)(122,217)(123,216)(124,215)(125,214)(126,213)(127,212)(128,211)(129,210)(130,209)(131,208)(132,207)(133,206)(134,205)(135,204)(136,203)(137,202)(138,201)(139,200)(140,199)(141,198)(142,197)(143,196)(144,195)(145,194)(146,193)(147,192)(148,191)(149,190)(150,189)(151,188)(152,187)(153,186)(154,185)(155,184)(156,183)(157,182)(158,181)(159,180)(160,179)(161,178)(162,177)(163,176)(164,175)(165,174)(166,173)(167,172)(168,171)(169,170)

Polycyclic presentation:

2^3,5,271,5|a^2=c,b^2=c,d^5=f,b^a=bc,d^a=d^4f^4,e^a=e^270,f^a=f^4

Conjugacy classes & character table

13553 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");;
gap> coll := SingleCollector(F, [2, 2, 2, 5, 271, 5]);;
gap> SetPower(coll, 1, F.3);;
gap> SetPower(coll, 2, F.3);;
gap> SetPower(coll, 4, F.6);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 4, 1, F.4^4*F.6^4);;
gap> SetConjugate(coll, 5, 1, F.5^270);;
gap> SetConjugate(coll, 6, 1, F.6^4);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic13550 #72289655