Dic₁₃₃₇₅ #72289131

order	53500
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₃₃₇₅

Splittings

derived, Hall	C₁₀₇×C₁₂₅ ⋊ C₄
Sylow	C₁₂₅ ⋊ Dic₁₀₇
Sylow	C₁₀₇ ⋊ Dic₁₂₅

Subgroups

	subgroup	G/H
center, hypercenter	C₂	D₁₃₃₇₅
commutator, hypocenter	C₁₀₇×C₁₂₅	C₄
Frattini	C₂×C₂₅	D₅₃₅
Fitting, cosocle	C₂×C₁₀₇×C₁₂₅	C₂
socle, prime radical	C₂×C₅×C₁₀₇	D₂₅
LCS	G ▷ C₁₀₇×C₁₂₅
UCS	C₁ ◁ C₂

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 236
transitive degree	μ_t(G) = 53500

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

Polycyclic presentation:

2^2,5,107,5^2|a^2=b,c^5=e,e^5=f,c^a=c^4e^4f^4,d^a=d^106,e^a=e^4f^4,f^a=f^4

Conjugacy classes & character table

13378 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, 5, 107, 5, 5]);;
gap> SetPower(coll, 1, F.2);;
gap> SetPower(coll, 3, F.5);;
gap> SetPower(coll, 5, F.6);;
gap> SetConjugate(coll, 3, 1, F.3^4*F.5^4*F.6^4);;
gap> SetConjugate(coll, 4, 1, F.4^106);;
gap> SetConjugate(coll, 5, 1, F.5^4*F.6^4);;
gap> SetConjugate(coll, 6, 1, F.6^4);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic13375 #72289131