Dic₁₀₃₀₂ #72274357

order	41208
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₀₃₀₂

Splittings

Hall	C₃×C₁₇×C₁₀₁ ⋊ Q₈
Hall	C₁₇×C₁₀₁ ⋊ Dic₆
Hall	C₃×C₁₀₁ ⋊ Dic₃₄
Sylow	C₁₀₁ ⋊ Dic₁₀₂
Hall	C₃×C₁₇ ⋊ Dic₂₀₂
Sylow	C₁₇ ⋊ Dic₆₀₆
Sylow	C₃ ⋊ Dic₃₄₃₄

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₅₁₅₁
commutator, socle, prime radical, cosocle	C₂×C₃×C₁₇×C₁₀₁	C₂²
Fitting	C₃×C₄×C₁₇×C₁₀₁	C₂
hypercenter	C₄	D₅₁₅₁
hypocenter	C₃×C₁₇×C₁₀₁	Q₈
LCS	G ▷ C₂×C₃×C₁₇×C₁₀₁ ▷ C₃×C₁₇×C₁₀₁
UCS	C₁ ◁ C₂ ◁ C₄

48 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 129
transitive degree	μ_t(G) = 41208

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

Polycyclic presentation:

2^3,3,17,101|a^2=c,b^2=c,b^a=bc,d^a=d^2,e^a=e^16,f^a=f^100

Conjugacy classes & character table

10305 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, 3, 17, 101]);;
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.5^16);;
gap> SetConjugate(coll, 6, 1, F.6^100);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic10302 #72274357