Dic₂₅₀₂ #72212765

order	10008
automorphism	Aut(G) order 8286624
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₂₅₀₂

Splittings

Hall	C₉×C₁₃₉ ⋊ Q₈
Sylow	C₁₃₉ ⋊ Dic₁₈
Sylow	C₉ ⋊ Dic₂₇₈

Subgroups

	subgroup	G/H
center	C₂	C₂×D₁₂₅₁
commutator, cosocle	C₂×C₉×C₁₃₉	C₂²
Frattini	C₂×C₃	C₂×D₄₁₇
Fitting	C₄×C₉×C₁₃₉	C₂
socle, prime radical	C₂×C₃×C₁₃₉	C₂×D₃
hypercenter	C₄	D₁₂₅₁
hypocenter	C₉×C₁₃₉	Q₈
LCS	G ▷ C₂×C₉×C₁₃₉ ▷ C₉×C₁₃₉
UCS	C₁ ◁ C₂ ◁ C₄

36 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 156
transitive degree	μ_t(G) = 10008

A minimal faithful permutation representation — 2 generators on 156 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);(1,5,3,7)(2,6,4,8)(10,17)(11,16)(12,15)(13,14)(19,156)(20,155)(21,154)(22,153)(23,152)(24,151)(25,150)(26,149)(27,148)(28,147)(29,146)(30,145)(31,144)(32,143)(33,142)(34,141)(35,140)(36,139)(37,138)(38,137)(39,136)(40,135)(41,134)(42,133)(43,132)(44,131)(45,130)(46,129)(47,128)(48,127)(49,126)(50,125)(51,124)(52,123)(53,122)(54,121)(55,120)(56,119)(57,118)(58,117)(59,116)(60,115)(61,114)(62,113)(63,112)(64,111)(65,110)(66,109)(67,108)(68,107)(69,106)(70,105)(71,104)(72,103)(73,102)(74,101)(75,100)(76,99)(77,98)(78,97)(79,96)(80,95)(81,94)(82,93)(83,92)(84,91)(85,90)(86,89)(87,88)

Polycyclic presentation:

2^3,3,139,3|a^2=c,b^2=c,d^3=f,b^a=bc,d^a=d^2f^2,e^a=e^138,f^a=f^2

Conjugacy classes & character table

2505 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₅₀₀₄, Dic₇₅₀₆, Dic₁₂₅₁₀
maximal normal subgroup of	Dic₅₀₀₄
maximal quotient of	Dic₇₅₀₆, Dic₁₂₅₁₀

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, 3, 139, 3]);;
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.4^2*F.6^2);;
gap> SetConjugate(coll, 5, 1, F.5^138);;
gap> SetConjugate(coll, 6, 1, F.6^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic2502 #72212765