Dic₄₀₈₇ #30654792

order	16348
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₄₀₈₇
unambiguous semidirect product	C₆₇⋊Dic₆₁

Splittings

derived, Hall	C₆₁×C₆₇ ⋊ C₄
Sylow	C₆₇ ⋊ Dic₆₁
Sylow	C₆₁ ⋊ Dic₆₇

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₄₀₈₇
commutator, hypocenter	C₆₁×C₆₇	C₄
Fitting, socle, prime radical, cosocle	C₂×C₆₁×C₆₇	C₂
LCS	G ▷ C₆₁×C₆₇
UCS	C₁ ◁ C₂

12 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 132
transitive degree	μ_t(G) = 16348

A minimal faithful permutation representation — 2 generators on 132 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);(1,3,2,4)(6,65)(7,64)(8,63)(9,62)(10,61)(11,60)(12,59)(13,58)(14,57)(15,56)(16,55)(17,54)(18,53)(19,52)(20,51)(21,50)(22,49)(23,48)(24,47)(25,46)(26,45)(27,44)(28,43)(29,42)(30,41)(31,40)(32,39)(33,38)(34,37)(35,36)(67,132)(68,131)(69,130)(70,129)(71,128)(72,127)(73,126)(74,125)(75,124)(76,123)(77,122)(78,121)(79,120)(80,119)(81,118)(82,117)(83,116)(84,115)(85,114)(86,113)(87,112)(88,111)(89,110)(90,109)(91,108)(92,107)(93,106)(94,105)(95,104)(96,103)(97,102)(98,101)(99,100)

Polycyclic presentation:

2,61,67,2|a^2=d,b^a=b^60,c^a=c^66

Conjugacy classes & character table

4090 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₈₁₇₄
maximal normal subgroup of	Dic₈₁₇₄

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(16348, 4);

Dic4087 #30654792