Dic₃₂₅₀ #72213626

order	13000
automorphism	Aut(G) order 15600000
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₃₂₅₀

Splittings

Hall	C₁₃×C₁₂₅ ⋊ Q₈
Sylow	C₁₂₅ ⋊ Dic₂₆
Sylow	C₁₃ ⋊ Dic₂₅₀

Subgroups

	subgroup	G/H
center	C₂	C₂×D₁₆₂₅
commutator, cosocle	C₂×C₁₃×C₁₂₅	C₂²
Frattini	C₂×C₂₅	C₂×D₆₅
Fitting	C₄×C₁₃×C₁₂₅	C₂
socle, prime radical	C₂×C₅×C₁₃	C₂×D₂₅
hypercenter	C₄	D₁₆₂₅
hypocenter	C₁₃×C₁₂₅	Q₈
LCS	G ▷ C₂×C₁₃×C₁₂₅ ▷ C₁₃×C₁₂₅
UCS	C₁ ◁ C₂ ◁ C₄

48 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 146
transitive degree	μ_t(G) = 13000

A minimal faithful permutation representation — 2 generators on 146 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);(1,5,3,7)(2,6,4,8)(10,133)(11,132)(12,131)(13,130)(14,129)(15,128)(16,127)(17,126)(18,125)(19,124)(20,123)(21,122)(22,121)(23,120)(24,119)(25,118)(26,117)(27,116)(28,115)(29,114)(30,113)(31,112)(32,111)(33,110)(34,109)(35,108)(36,107)(37,106)(38,105)(39,104)(40,103)(41,102)(42,101)(43,100)(44,99)(45,98)(46,97)(47,96)(48,95)(49,94)(50,93)(51,92)(52,91)(53,90)(54,89)(55,88)(56,87)(57,86)(58,85)(59,84)(60,83)(61,82)(62,81)(63,80)(64,79)(65,78)(66,77)(67,76)(68,75)(69,74)(70,73)(71,72)(135,146)(136,145)(137,144)(138,143)(139,142)(140,141)

Polycyclic presentation:

2^3,5,13,5^2|a^2=c,b^2=c,d^5=f,f^5=g,b^a=bc,d^a=d^4f^4g^4,e^a=e^12,f^a=f^4g^4,g^a=g^4

Conjugacy classes & character table

3253 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₆₅₀₀, Dic₉₇₅₀, Dic₁₆₂₅₀
maximal normal subgroup of	Dic₆₅₀₀
maximal quotient of	Dic₉₇₅₀, Dic₁₆₂₅₀

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, 2, 5, 13, 5, 5]);;
gap> SetPower(coll, 1, F.3);;
gap> SetPower(coll, 2, F.3);;
gap> SetPower(coll, 4, F.6);;
gap> SetPower(coll, 6, F.7);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 4, 1, F.4^4*F.6^4*F.7^4);;
gap> SetConjugate(coll, 5, 1, F.5^12);;
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);

Dic3250 #72213626