Dic₄₁₂₅ #72215474

order	16500
automorphism	Aut(G) order 16500000
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₄₁₂₅

Splittings

derived, Hall	C₃×C₁₁×C₁₂₅ ⋊ C₄
Hall	C₁₁×C₁₂₅ ⋊ Dic₃
Hall	C₃×C₁₂₅ ⋊ Dic₁₁
Sylow	C₁₂₅ ⋊ Dic₃₃
Hall	C₃×C₁₁ ⋊ Dic₁₂₅
Sylow	C₁₁ ⋊ Dic₃₇₅
Sylow	C₃ ⋊ Dic₁₃₇₅

Subgroups

	subgroup	G/H
center, hypercenter	C₂	D₄₁₂₅
commutator, hypocenter	C₃×C₁₁×C₁₂₅	C₄
Frattini	C₂×C₂₅	D₁₆₅
Fitting, cosocle	C₂×C₃×C₁₁×C₁₂₅	C₂
socle, prime radical	C₂×C₃×C₅×C₁₁	D₂₅
LCS	G ▷ C₃×C₁₁×C₁₂₅
UCS	C₁ ◁ C₂

48 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 143
transitive degree	μ_t(G) = 16500

A minimal faithful permutation representation — 2 generators on 143 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);(1,3,2,4)(6,7)(9,132)(10,131)(11,130)(12,129)(13,128)(14,127)(15,126)(16,125)(17,124)(18,123)(19,122)(20,121)(21,120)(22,119)(23,118)(24,117)(25,116)(26,115)(27,114)(28,113)(29,112)(30,111)(31,110)(32,109)(33,108)(34,107)(35,106)(36,105)(37,104)(38,103)(39,102)(40,101)(41,100)(42,99)(43,98)(44,97)(45,96)(46,95)(47,94)(48,93)(49,92)(50,91)(51,90)(52,89)(53,88)(54,87)(55,86)(56,85)(57,84)(58,83)(59,82)(60,81)(61,80)(62,79)(63,78)(64,77)(65,76)(66,75)(67,74)(68,73)(69,72)(70,71)(134,143)(135,142)(136,141)(137,140)(138,139)

Polycyclic presentation:

2^2,3,5,11,5^2|a^2=b,d^5=f,f^5=g,c^a=c^2,d^a=d^4f^4g^4,e^a=e^10,f^a=f^4g^4,g^a=g^4

Conjugacy classes & character table

4128 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₈₂₅₀, Dic₁₂₃₇₅
maximal normal subgroup of	Dic₈₂₅₀
maximal quotient of	Dic₁₂₃₇₅

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, 3, 5, 11, 5, 5]);;
gap> SetPower(coll, 1, F.2);;
gap> SetPower(coll, 4, F.6);;
gap> SetPower(coll, 6, F.7);;
gap> SetConjugate(coll, 3, 1, F.3^2);;
gap> SetConjugate(coll, 4, 1, F.4^4*F.6^4*F.7^4);;
gap> SetConjugate(coll, 5, 1, F.5^10);;
gap> SetConjugate(coll, 6, 1, F.6^4*F.7^4);;
gap> SetConjugate(coll, 7, 1, F.7^4);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic4125 #72215474