Dic₁₅₆₅₆ #72295413

order	62624
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₅₆₅₆

Splittings

Hall	C₁₉×C₁₀₃ ⋊ Q₃₂
Sylow	C₁₀₃ ⋊ Dic₁₅₂
Sylow	C₁₉ ⋊ Dic₈₂₄

Subgroups

	subgroup	G/H
center	C₂	D₁₅₆₅₆
commutator, cosocle	C₈×C₁₉×C₁₀₃	C₂²
Frattini	C₈	C₂×D₁₉₅₇
Fitting	C₁₆×C₁₉×C₁₀₃	C₂
socle, prime radical	C₂×C₁₉×C₁₀₃	D₈
hypercenter	C₁₆	D₁₉₅₇
hypocenter	C₁₉×C₁₀₃	Q₃₂
LCS	G ▷ C₈×C₁₉×C₁₀₃ ▷ C₄×C₁₉×C₁₀₃ ▷ C₂×C₁₉×C₁₀₃ ▷ C₁₉×C₁₀₃
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆

48 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 154
transitive degree	μ_t(G) = 62624

A minimal faithful permutation representation — 2 generators on 154 points:

(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16)(17,32,31,30,29,28,27,26,25,24,23,22,21,20,19,18)(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);(1,17,9,25)(2,18,10,26)(3,19,11,27)(4,20,12,28)(5,21,13,29)(6,22,14,30)(7,23,15,31)(8,24,16,32)(34,51)(35,50)(36,49)(37,48)(38,47)(39,46)(40,45)(41,44)(42,43)(53,154)(54,153)(55,152)(56,151)(57,150)(58,149)(59,148)(60,147)(61,146)(62,145)(63,144)(64,143)(65,142)(66,141)(67,140)(68,139)(69,138)(70,137)(71,136)(72,135)(73,134)(74,133)(75,132)(76,131)(77,130)(78,129)(79,128)(80,127)(81,126)(82,125)(83,124)(84,123)(85,122)(86,121)(87,120)(88,119)(89,118)(90,117)(91,116)(92,115)(93,114)(94,113)(95,112)(96,111)(97,110)(98,109)(99,108)(100,107)(101,106)(102,105)(103,104)

Polycyclic presentation:

2^5,19,103|a^2=e,b^2=cd,c^2=de,d^2=e,b^a=bc,c^a=cd,d^a=de,f^a=f^18,g^a=g^102

Conjugacy classes & character table

15659 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", "g");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 19, 103]);;
gap> SetPower(coll, 1, F.5);;
gap> SetPower(coll, 2, F.3*F.4);;
gap> SetPower(coll, 3, F.4*F.5);;
gap> SetPower(coll, 4, F.5);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 4, 1, F.4*F.5);;
gap> SetConjugate(coll, 6, 1, F.6^18);;
gap> SetConjugate(coll, 7, 1, F.7^102);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic15656 #72295413