Dic₄₁₇₀ #72215539

order	16680
automorphism	Aut(G) order 18414720
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₄₁₇₀

Splittings

Hall	C₃×C₅×C₁₃₉ ⋊ Q₈
Hall	C₅×C₁₃₉ ⋊ Dic₆
Hall	C₃×C₁₃₉ ⋊ Dic₁₀
Sylow	C₁₃₉ ⋊ Dic₃₀
Hall	C₃×C₅ ⋊ Dic₂₇₈
Sylow	C₅ ⋊ Dic₈₃₄
Sylow	C₃ ⋊ Dic₁₃₉₀

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₂₀₈₅
commutator, socle, prime radical, cosocle	C₂×C₃×C₅×C₁₃₉	C₂²
Fitting	C₃×C₄×C₅×C₁₃₉	C₂
hypercenter	C₄	D₂₀₈₅
hypocenter	C₃×C₅×C₁₃₉	Q₈
LCS	G ▷ C₂×C₃×C₅×C₁₃₉ ▷ C₃×C₅×C₁₃₉
UCS	C₁ ◁ C₂ ◁ C₄

48 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 155
transitive degree	μ_t(G) = 16680

A minimal faithful permutation representation — 2 generators on 155 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);(1,5,3,7)(2,6,4,8)(10,11)(13,16)(14,15)(18,155)(19,154)(20,153)(21,152)(22,151)(23,150)(24,149)(25,148)(26,147)(27,146)(28,145)(29,144)(30,143)(31,142)(32,141)(33,140)(34,139)(35,138)(36,137)(37,136)(38,135)(39,134)(40,133)(41,132)(42,131)(43,130)(44,129)(45,128)(46,127)(47,126)(48,125)(49,124)(50,123)(51,122)(52,121)(53,120)(54,119)(55,118)(56,117)(57,116)(58,115)(59,114)(60,113)(61,112)(62,111)(63,110)(64,109)(65,108)(66,107)(67,106)(68,105)(69,104)(70,103)(71,102)(72,101)(73,100)(74,99)(75,98)(76,97)(77,96)(78,95)(79,94)(80,93)(81,92)(82,91)(83,90)(84,89)(85,88)(86,87)

Polycyclic presentation:

2^3,3,5,139|a^2=c,b^2=c,b^a=bc,d^a=d^2,e^a=e^4,f^a=f^138

Conjugacy classes & character table

4173 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₈₃₄₀, Dic₁₂₅₁₀
maximal normal subgroup of	Dic₈₃₄₀
maximal quotient of	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, 5, 139]);;
gap> SetPower(coll, 1, F.3);;
gap> SetPower(coll, 2, F.3);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 4, 1, F.4^2);;
gap> SetConjugate(coll, 5, 1, F.5^4);;
gap> SetConjugate(coll, 6, 1, F.6^138);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic4170 #72215539