Dic₄₁₂ #13110096

order	1648
automorphism	Aut(G) order 336192
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₄₁₂

Splittings

Sylow

C₁₀₃ ⋊ Q₁₆

Subgroups

	subgroup	G/H
center	C₂	D₄₁₂
commutator, cosocle	C₄×C₁₀₃	C₂²
Frattini	C₄	C₂×D₁₀₃
Fitting	C₈×C₁₀₃	C₂
socle, prime radical	C₂×C₁₀₃	D₄
hypercenter	C₈	D₁₀₃
hypocenter	C₁₀₃	Q₁₆
LCS	G ▷ C₄×C₁₀₃ ▷ C₂×C₁₀₃ ▷ C₁₀₃
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

18 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 119
transitive degree	μ_t(G) = 1648

A minimal faithful permutation representation — 2 generators on 119 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);(1,9,5,13)(2,10,6,14)(3,11,7,15)(4,12,8,16)(18,119)(19,118)(20,117)(21,116)(22,115)(23,114)(24,113)(25,112)(26,111)(27,110)(28,109)(29,108)(30,107)(31,106)(32,105)(33,104)(34,103)(35,102)(36,101)(37,100)(38,99)(39,98)(40,97)(41,96)(42,95)(43,94)(44,93)(45,92)(46,91)(47,90)(48,89)(49,88)(50,87)(51,86)(52,85)(53,84)(54,83)(55,82)(56,81)(57,80)(58,79)(59,78)(60,77)(61,76)(62,75)(63,74)(64,73)(65,72)(66,71)(67,70)(68,69)

Polycyclic presentation:

2^4,103|a^2=d,b^2=d,c^2=d,b^a=bc,c^a=cd,e^a=e^102,c^b=cd,e^b=e^102

Conjugacy classes & character table

415 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#15692965, #15692966, #15692967, …
maximal normal subgroup of	#15692965, #15692966, #15692967, …
maximal quotient of	#15692946, #15692955, C₂×Dic₄₁₂, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1648, 31);

Dic412 #13110096