Dic₁₄₀₆₅ #72290485

order	56260
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₄₀₆₅

Splittings

derived, Hall	C₅×C₂₉×C₉₇ ⋊ C₄
Hall	C₂₉×C₉₇ ⋊ Dic₅
Hall	C₅×C₉₇ ⋊ Dic₂₉
Hall	C₅×C₂₉ ⋊ Dic₉₇
Sylow	C₉₇ ⋊ Dic₁₄₅
Sylow	C₂₉ ⋊ Dic₄₈₅
Sylow	C₅ ⋊ Dic₂₈₁₃

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₁₄₀₆₅
commutator, hypocenter	C₅×C₂₉×C₉₇	C₄
Fitting, socle, prime radical, cosocle	C₂×C₅×C₂₉×C₉₇	C₂
LCS	G ▷ C₅×C₂₉×C₉₇
UCS	C₁ ◁ C₂

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 135
transitive degree	μ_t(G) = 56260

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)(6,9)(7,8)(11,38)(12,37)(13,36)(14,35)(15,34)(16,33)(17,32)(18,31)(19,30)(20,29)(21,28)(22,27)(23,26)(24,25)(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:

2^2,5,29,97|a^2=b,c^a=c^4,d^a=d^28,e^a=e^96

Conjugacy classes & character table

14068 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, 5, 29, 97]);;
gap> SetPower(coll, 1, F.2);;
gap> SetConjugate(coll, 3, 1, F.3^4);;
gap> SetConjugate(coll, 4, 1, F.4^28);;
gap> SetConjugate(coll, 5, 1, F.5^96);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic14065 #72290485