Dic₆₀₇₅ #72224302

order	24300
automorphism	Aut(G) order 39366000
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₆₀₇₅

Splittings

derived, Hall	C₂₅×C₂₄₃ ⋊ C₄
Sylow	C₂₄₃ ⋊ Dic₂₅
Sylow	C₂₅ ⋊ Dic₂₄₃

Subgroups

	subgroup	G/H
center, hypercenter	C₂	D₆₀₇₅
commutator, hypocenter	C₂₅×C₂₄₃	C₄
Frattini	C₂×C₅×C₈₁	D₁₅
Fitting, cosocle	C₂×C₂₅×C₂₄₃	C₂
socle, prime radical	C₂×C₃×C₅	D₄₀₅
LCS	G ▷ C₂₅×C₂₄₃
UCS	C₁ ◁ C₂

54 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 272
transitive degree	μ_t(G) = 24300

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

Polycyclic presentation:

2^2,3,5,3,5,3^3|a^2=b,c^3=e,d^5=f,e^3=g,g^3=h,h^3=i,c^a=c^2e^2g^2h^2i^2,d^a=d^4f^4,e^a=e^2g^2h^2i^2,f^a=f^4,g^a=g^2h^2i^2,h^a=h^2i^2,i^a=i^2

Conjugacy classes & character table

6078 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₁₂₁₅₀
maximal normal subgroup of	Dic₁₂₁₅₀

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", "h", "i");;
gap> coll := SingleCollector(F, [2, 2, 3, 5, 3, 5, 3, 3, 3]);;
gap> SetPower(coll, 1, F.2);;
gap> SetPower(coll, 3, F.5);;
gap> SetPower(coll, 4, F.6);;
gap> SetPower(coll, 5, F.7);;
gap> SetPower(coll, 7, F.8);;
gap> SetPower(coll, 8, F.9);;
gap> SetConjugate(coll, 3, 1, F.3^2*F.5^2*F.7^2*F.8^2*F.9^2);;
gap> SetConjugate(coll, 4, 1, F.4^4*F.6^4);;
gap> SetConjugate(coll, 5, 1, F.5^2*F.7^2*F.8^2*F.9^2);;
gap> SetConjugate(coll, 6, 1, F.6^4);;
gap> SetConjugate(coll, 7, 1, F.7^2*F.8^2*F.9^2);;
gap> SetConjugate(coll, 8, 1, F.8^2*F.9^2);;
gap> SetConjugate(coll, 9, 1, F.9^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic6075 #72224302