Dic₁₉₅₆ #72212331

order	7824
automorphism	Aut(G) order 5069952
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₉₅₆

Splittings

Hall	C₃×C₁₆₃ ⋊ Q₁₆
Sylow	C₁₆₃ ⋊ Dic₁₂
Sylow	C₃ ⋊ Dic₆₅₂

Subgroups

	subgroup	G/H
center	C₂	D₁₉₅₆
commutator, cosocle	C₃×C₄×C₁₆₃	C₂²
Frattini	C₄	C₂×D₄₈₉
Fitting	C₃×C₈×C₁₆₃	C₂
socle, prime radical	C₂×C₃×C₁₆₃	D₄
hypercenter	C₈	D₄₈₉
hypocenter	C₃×C₁₆₃	Q₁₆
LCS	G ▷ C₃×C₄×C₁₆₃ ▷ C₂×C₃×C₁₆₃ ▷ C₃×C₁₆₃
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

36 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 182
transitive degree	μ_t(G) = 7824

A minimal faithful permutation representation — 2 generators on 182 points:

(1,2,3,4,5,6,7,8)(9,16,15,14,13,12,11,10)(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);(1,9,5,13)(2,10,6,14)(3,11,7,15)(4,12,8,16)(18,19)(21,182)(22,181)(23,180)(24,179)(25,178)(26,177)(27,176)(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:

2^4,3,163|a^2=d,b^2=cd,c^2=d,b^a=bc,c^a=cd,e^a=e^2,f^a=f^162

Conjugacy classes & character table

1959 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₃₉₁₂, Dic₅₈₆₈, Dic₉₇₈₀, …
maximal normal subgroup of	Dic₃₉₁₂
maximal quotient of	Dic₅₈₆₈, Dic₉₇₈₀, Dic₁₃₆₉₂

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, 2, 2, 3, 163]);;
gap> SetPower(coll, 1, F.4);;
gap> SetPower(coll, 2, F.3*F.4);;
gap> SetPower(coll, 3, F.4);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 5, 1, F.5^2);;
gap> SetConjugate(coll, 6, 1, F.6^162);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic1956 #72212331