Dic₁₅₃₀₁ #72293209

order	61204
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₅₃₀₁

Splittings

derived, Hall	C₁₁×C₁₃×C₁₀₇ ⋊ C₄
Hall	C₁₃×C₁₀₇ ⋊ Dic₁₁
Hall	C₁₁×C₁₀₇ ⋊ Dic₁₃
Hall	C₁₁×C₁₃ ⋊ Dic₁₀₇
Sylow	C₁₀₇ ⋊ Dic₁₄₃
Sylow	C₁₃ ⋊ Dic₁₁₇₇
Sylow	C₁₁ ⋊ Dic₁₃₉₁

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₁₅₃₀₁
commutator, hypocenter	C₁₁×C₁₃×C₁₀₇	C₄
Fitting, socle, prime radical, cosocle	C₂×C₁₁×C₁₃×C₁₀₇	C₂
LCS	G ▷ C₁₁×C₁₃×C₁₀₇
UCS	C₁ ◁ C₂

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 135
transitive degree	μ_t(G) = 61204

A minimal faithful permutation representation — 2 generators on 135 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);(1,3,2,4)(6,15)(7,14)(8,13)(9,12)(10,11)(17,28)(18,27)(19,26)(20,25)(21,24)(22,23)(30,135)(31,134)(32,133)(33,132)(34,131)(35,130)(36,129)(37,128)(38,127)(39,126)(40,125)(41,124)(42,123)(43,122)(44,121)(45,120)(46,119)(47,118)(48,117)(49,116)(50,115)(51,114)(52,113)(53,112)(54,111)(55,110)(56,109)(57,108)(58,107)(59,106)(60,105)(61,104)(62,103)(63,102)(64,101)(65,100)(66,99)(67,98)(68,97)(69,96)(70,95)(71,94)(72,93)(73,92)(74,91)(75,90)(76,89)(77,88)(78,87)(79,86)(80,85)(81,84)(82,83)

Polycyclic presentation:

2^2,11,13,107|a^2=b,c^a=c^10,d^a=d^12,e^a=e^106

Conjugacy classes & character table

15304 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");;
gap> coll := SingleCollector(F, [2, 2, 11, 13, 107]);;
gap> SetPower(coll, 1, F.2);;
gap> SetConjugate(coll, 3, 1, F.3^10);;
gap> SetConjugate(coll, 4, 1, F.4^12);;
gap> SetConjugate(coll, 5, 1, F.5^106);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic15301 #72293209