Dic₁₄₁₇₀ #72290657

order	56680
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₄₁₇₀

Splittings

Hall	C₅×C₁₃×C₁₀₉ ⋊ Q₈
Hall	C₁₃×C₁₀₉ ⋊ Dic₁₀
Hall	C₅×C₁₀₉ ⋊ Dic₂₆
Sylow	C₁₀₉ ⋊ Dic₁₃₀
Hall	C₅×C₁₃ ⋊ Dic₂₁₈
Sylow	C₁₃ ⋊ Dic₁₀₉₀
Sylow	C₅ ⋊ Dic₂₈₃₄

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₇₀₈₅
commutator, socle, prime radical, cosocle	C₂×C₅×C₁₃×C₁₀₉	C₂²
Fitting	C₄×C₅×C₁₃×C₁₀₉	C₂
hypercenter	C₄	D₇₀₈₅
hypocenter	C₅×C₁₃×C₁₀₉	Q₈
LCS	G ▷ C₂×C₅×C₁₃×C₁₀₉ ▷ C₅×C₁₃×C₁₀₉
UCS	C₁ ◁ C₂ ◁ C₄

48 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 135
transitive degree	μ_t(G) = 56680

A minimal faithful permutation representation — 2 generators on 135 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);(1,5,3,7)(2,6,4,8)(10,13)(11,12)(15,26)(16,25)(17,24)(18,23)(19,22)(20,21)(28,135)(29,134)(30,133)(31,132)(32,131)(33,130)(34,129)(35,128)(36,127)(37,126)(38,125)(39,124)(40,123)(41,122)(42,121)(43,120)(44,119)(45,118)(46,117)(47,116)(48,115)(49,114)(50,113)(51,112)(52,111)(53,110)(54,109)(55,108)(56,107)(57,106)(58,105)(59,104)(60,103)(61,102)(62,101)(63,100)(64,99)(65,98)(66,97)(67,96)(68,95)(69,94)(70,93)(71,92)(72,91)(73,90)(74,89)(75,88)(76,87)(77,86)(78,85)(79,84)(80,83)(81,82)

Polycyclic presentation:

2^3,5,13,109|a^2=c,b^2=c,b^a=bc,d^a=d^4,e^a=e^12,f^a=f^108

Conjugacy classes & character table

14173 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, 2, 5, 13, 109]);;
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.4^4);;
gap> SetConjugate(coll, 5, 1, F.5^12);;
gap> SetConjugate(coll, 6, 1, F.6^108);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic14170 #72290657