Dic₉₆₈ #72211533

order	3872
automorphism	Aut(G) order 1703680
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₉₆₈

Splittings

Sylow

C₁₂₁ ⋊ Q₃₂

Subgroups

	subgroup	G/H
center	C₂	D₉₆₈
commutator, cosocle	C₈×C₁₂₁	C₂²
Frattini	C₈×C₁₁	C₂×D₁₁
Fitting	C₁₆×C₁₂₁	C₂
socle, prime radical	C₂×C₁₁	D₈₈
hypercenter	C₁₆	D₁₂₁
hypocenter	C₁₂₁	Q₃₂
LCS	G ▷ C₈×C₁₂₁ ▷ C₄×C₁₂₁ ▷ C₂×C₁₂₁ ▷ C₁₂₁
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆

36 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 153
transitive degree	μ_t(G) = 3872

A minimal faithful permutation representation — 2 generators on 153 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);(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)(34,153)(35,152)(36,151)(37,150)(38,149)(39,148)(40,147)(41,146)(42,145)(43,144)(44,143)(45,142)(46,141)(47,140)(48,139)(49,138)(50,137)(51,136)(52,135)(53,134)(54,133)(55,132)(56,131)(57,130)(58,129)(59,128)(60,127)(61,126)(62,125)(63,124)(64,123)(65,122)(66,121)(67,120)(68,119)(69,118)(70,117)(71,116)(72,115)(73,114)(74,113)(75,112)(76,111)(77,110)(78,109)(79,108)(80,107)(81,106)(82,105)(83,104)(84,103)(85,102)(86,101)(87,100)(88,99)(89,98)(90,97)(91,96)(92,95)(93,94)

Polycyclic presentation:

2^5,11^2|a^2=e,b^2=cd,c^2=de,d^2=e,f^11=g,b^a=bc,c^a=cd,d^a=de,f^a=f^10g^10,g^a=g^10

Conjugacy classes & character table

971 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	Dic₁₉₃₆, Dic₂₉₀₄, Dic₄₈₄₀, …
maximal normal subgroup of	Dic₁₉₃₆
maximal quotient of	Dic₂₉₀₄, Dic₄₈₄₀, Dic₆₇₇₆, …

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, 11, 11]);;
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> SetPower(coll, 6, F.7);;
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.6^10*F.7^10);;
gap> SetConjugate(coll, 7, 1, F.7^10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic968 #72211533