Dic₁₆₀₂₃ #72298288

order	64092
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₆₀₂₃

Splittings

derived, Hall	C₃×C₄₉×C₁₀₉ ⋊ C₄
Hall	C₄₉×C₁₀₉ ⋊ Dic₃
Hall	C₃×C₁₀₉ ⋊ Dic₄₉
Hall	C₃×C₄₉ ⋊ Dic₁₀₉
Sylow	C₁₀₉ ⋊ Dic₁₄₇
Sylow	C₄₉ ⋊ Dic₃₂₇
Sylow	C₃ ⋊ Dic₅₃₄₁

Subgroups

	subgroup	G/H
center, hypercenter	C₂	D₁₆₀₂₃
commutator, hypocenter	C₃×C₄₉×C₁₀₉	C₄
Frattini	C₂×C₇	D₂₂₈₉
Fitting, cosocle	C₂×C₃×C₄₉×C₁₀₉	C₂
socle, prime radical	C₂×C₃×C₇×C₁₀₉	D₇
LCS	G ▷ C₃×C₄₉×C₁₀₉
UCS	C₁ ◁ C₂

36 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 165
transitive degree	μ_t(G) = 64092

A minimal faithful permutation representation — 2 generators on 165 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);(1,3,2,4)(6,7)(9,56)(10,55)(11,54)(12,53)(13,52)(14,51)(15,50)(16,49)(17,48)(18,47)(19,46)(20,45)(21,44)(22,43)(23,42)(24,41)(25,40)(26,39)(27,38)(28,37)(29,36)(30,35)(31,34)(32,33)(58,165)(59,164)(60,163)(61,162)(62,161)(63,160)(64,159)(65,158)(66,157)(67,156)(68,155)(69,154)(70,153)(71,152)(72,151)(73,150)(74,149)(75,148)(76,147)(77,146)(78,145)(79,144)(80,143)(81,142)(82,141)(83,140)(84,139)(85,138)(86,137)(87,136)(88,135)(89,134)(90,133)(91,132)(92,131)(93,130)(94,129)(95,128)(96,127)(97,126)(98,125)(99,124)(100,123)(101,122)(102,121)(103,120)(104,119)(105,118)(106,117)(107,116)(108,115)(109,114)(110,113)(111,112)

Polycyclic presentation:

2^2,3,7,109,7|a^2=b,d^7=f,c^a=c^2,d^a=d^6f^6,e^a=e^108,f^a=f^6

Conjugacy classes & character table

16026 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, 3, 7, 109, 7]);;
gap> SetPower(coll, 1, F.2);;
gap> SetPower(coll, 4, F.6);;
gap> SetConjugate(coll, 3, 1, F.3^2);;
gap> SetConjugate(coll, 4, 1, F.4^6*F.6^6);;
gap> SetConjugate(coll, 5, 1, F.5^108);;
gap> SetConjugate(coll, 6, 1, F.6^6);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic16023 #72298288