Dic₁₂₆₆₅ #72288024

order	50660
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₂₆₆₅

Splittings

derived, Hall	C₅×C₁₇×C₁₄₉ ⋊ C₄
Hall	C₁₇×C₁₄₉ ⋊ Dic₅
Hall	C₅×C₁₄₉ ⋊ Dic₁₇
Sylow	C₁₄₉ ⋊ Dic₈₅
Hall	C₅×C₁₇ ⋊ Dic₁₄₉
Sylow	C₁₇ ⋊ Dic₇₄₅
Sylow	C₅ ⋊ Dic₂₅₃₃

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₁₂₆₆₅
commutator, hypocenter	C₅×C₁₇×C₁₄₉	C₄
Fitting, socle, prime radical, cosocle	C₂×C₅×C₁₇×C₁₄₉	C₂
LCS	G ▷ C₅×C₁₇×C₁₄₉
UCS	C₁ ◁ C₂

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 175
transitive degree	μ_t(G) = 50660

A minimal faithful permutation representation — 2 generators on 175 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)(27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175);(1,3,2,4)(6,9)(7,8)(11,26)(12,25)(13,24)(14,23)(15,22)(16,21)(17,20)(18,19)(28,175)(29,174)(30,173)(31,172)(32,171)(33,170)(34,169)(35,168)(36,167)(37,166)(38,165)(39,164)(40,163)(41,162)(42,161)(43,160)(44,159)(45,158)(46,157)(47,156)(48,155)(49,154)(50,153)(51,152)(52,151)(53,150)(54,149)(55,148)(56,147)(57,146)(58,145)(59,144)(60,143)(61,142)(62,141)(63,140)(64,139)(65,138)(66,137)(67,136)(68,135)(69,134)(70,133)(71,132)(72,131)(73,130)(74,129)(75,128)(76,127)(77,126)(78,125)(79,124)(80,123)(81,122)(82,121)(83,120)(84,119)(85,118)(86,117)(87,116)(88,115)(89,114)(90,113)(91,112)(92,111)(93,110)(94,109)(95,108)(96,107)(97,106)(98,105)(99,104)(100,103)(101,102)

Polycyclic presentation:

2^2,5,17,149|a^2=b,c^a=c^4,d^a=d^16,e^a=e^148

Conjugacy classes & character table

12668 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, 17, 149]);;
gap> SetPower(coll, 1, F.2);;
gap> SetConjugate(coll, 3, 1, F.3^4);;
gap> SetConjugate(coll, 4, 1, F.4^16);;
gap> SetConjugate(coll, 5, 1, F.5^148);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic12665 #72288024