Dic₁₂₈₅₉ #72288307

order	51436
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₂₈₅₉

Splittings

derived, Hall	C₇×C₁₁×C₁₆₇ ⋊ C₄
Hall	C₁₁×C₁₆₇ ⋊ Dic₇
Hall	C₇×C₁₆₇ ⋊ Dic₁₁
Sylow	C₁₆₇ ⋊ Dic₇₇
Hall	C₇×C₁₁ ⋊ Dic₁₆₇
Sylow	C₁₁ ⋊ Dic₁₁₆₉
Sylow	C₇ ⋊ Dic₁₈₃₇

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₁₂₈₅₉
commutator, hypocenter	C₇×C₁₁×C₁₆₇	C₄
Fitting, socle, prime radical, cosocle	C₂×C₇×C₁₁×C₁₆₇	C₂
LCS	G ▷ C₇×C₁₁×C₁₆₇
UCS	C₁ ◁ C₂

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 189
transitive degree	μ_t(G) = 51436

A minimal faithful permutation representation — 2 generators on 189 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);(1,3,2,4)(6,11)(7,10)(8,9)(13,22)(14,21)(15,20)(16,19)(17,18)(24,189)(25,188)(26,187)(27,186)(28,185)(29,184)(30,183)(31,182)(32,181)(33,180)(34,179)(35,178)(36,177)(37,176)(38,175)(39,174)(40,173)(41,172)(42,171)(43,170)(44,169)(45,168)(46,167)(47,166)(48,165)(49,164)(50,163)(51,162)(52,161)(53,160)(54,159)(55,158)(56,157)(57,156)(58,155)(59,154)(60,153)(61,152)(62,151)(63,150)(64,149)(65,148)(66,147)(67,146)(68,145)(69,144)(70,143)(71,142)(72,141)(73,140)(74,139)(75,138)(76,137)(77,136)(78,135)(79,134)(80,133)(81,132)(82,131)(83,130)(84,129)(85,128)(86,127)(87,126)(88,125)(89,124)(90,123)(91,122)(92,121)(93,120)(94,119)(95,118)(96,117)(97,116)(98,115)(99,114)(100,113)(101,112)(102,111)(103,110)(104,109)(105,108)(106,107)

Polycyclic presentation:

2^2,7,11,167|a^2=b,c^a=c^6,d^a=d^10,e^a=e^166

Conjugacy classes & character table

12862 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, 7, 11, 167]);;
gap> SetPower(coll, 1, F.2);;
gap> SetConjugate(coll, 3, 1, F.3^6);;
gap> SetConjugate(coll, 4, 1, F.4^10);;
gap> SetConjugate(coll, 5, 1, F.5^166);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic12859 #72288307