Dic₁₅₄₈₅ #72293641

order	61940
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₅₄₈₅

Splittings

derived, Hall	C₅×C₁₉×C₁₆₃ ⋊ C₄
Hall	C₁₉×C₁₆₃ ⋊ Dic₅
Hall	C₅×C₁₆₃ ⋊ Dic₁₉
Sylow	C₁₆₃ ⋊ Dic₉₅
Hall	C₅×C₁₉ ⋊ Dic₁₆₃
Sylow	C₁₉ ⋊ Dic₈₁₅
Sylow	C₅ ⋊ Dic₃₀₉₇

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₁₅₄₈₅
commutator, hypocenter	C₅×C₁₉×C₁₆₃	C₄
Fitting, socle, prime radical, cosocle	C₂×C₅×C₁₉×C₁₆₃	C₂
LCS	G ▷ C₅×C₁₉×C₁₆₃
UCS	C₁ ◁ C₂

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 191
transitive degree	μ_t(G) = 61940

A minimal faithful permutation representation — 2 generators on 191 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)(29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191);(1,3,2,4)(6,9)(7,8)(11,28)(12,27)(13,26)(14,25)(15,24)(16,23)(17,22)(18,21)(19,20)(30,191)(31,190)(32,189)(33,188)(34,187)(35,186)(36,185)(37,184)(38,183)(39,182)(40,181)(41,180)(42,179)(43,178)(44,177)(45,176)(46,175)(47,174)(48,173)(49,172)(50,171)(51,170)(52,169)(53,168)(54,167)(55,166)(56,165)(57,164)(58,163)(59,162)(60,161)(61,160)(62,159)(63,158)(64,157)(65,156)(66,155)(67,154)(68,153)(69,152)(70,151)(71,150)(72,149)(73,148)(74,147)(75,146)(76,145)(77,144)(78,143)(79,142)(80,141)(81,140)(82,139)(83,138)(84,137)(85,136)(86,135)(87,134)(88,133)(89,132)(90,131)(91,130)(92,129)(93,128)(94,127)(95,126)(96,125)(97,124)(98,123)(99,122)(100,121)(101,120)(102,119)(103,118)(104,117)(105,116)(106,115)(107,114)(108,113)(109,112)(110,111)

Polycyclic presentation:

2^2,5,19,163|a^2=b,c^a=c^4,d^a=d^18,e^a=e^162

Conjugacy classes & character table

15488 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, 19, 163]);;
gap> SetPower(coll, 1, F.2);;
gap> SetConjugate(coll, 3, 1, F.3^4);;
gap> SetConjugate(coll, 4, 1, F.4^18);;
gap> SetConjugate(coll, 5, 1, F.5^162);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic15485 #72293641