Dic₄₂₈ #13133809

order	1712
automorphism	Aut(G) order 362944
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₄₂₈

Splittings

Sylow

C₁₀₇ ⋊ Q₁₆

Subgroups

	subgroup	G/H
center	C₂	D₄₂₈
commutator, cosocle	C₄×C₁₀₇	C₂²
Frattini	C₄	C₂×D₁₀₇
Fitting	C₈×C₁₀₇	C₂
socle, prime radical	C₂×C₁₀₇	D₄
hypercenter	C₈	D₁₀₇
hypocenter	C₁₀₇	Q₁₆
LCS	G ▷ C₄×C₁₀₇ ▷ C₂×C₁₀₇ ▷ C₁₀₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

18 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 123
transitive degree	μ_t(G) = 1712

A minimal faithful permutation representation — 2 generators on 123 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);(1,9,5,13)(2,10,6,14)(3,11,7,15)(4,12,8,16)(18,123)(19,122)(20,121)(21,120)(22,119)(23,118)(24,117)(25,116)(26,115)(27,114)(28,113)(29,112)(30,111)(31,110)(32,109)(33,108)(34,107)(35,106)(36,105)(37,104)(38,103)(39,102)(40,101)(41,100)(42,99)(43,98)(44,97)(45,96)(46,95)(47,94)(48,93)(49,92)(50,91)(51,90)(52,89)(53,88)(54,87)(55,86)(56,85)(57,84)(58,83)(59,82)(60,81)(61,80)(62,79)(63,78)(64,77)(65,76)(66,75)(67,74)(68,73)(69,72)(70,71)

Polycyclic presentation:

2^4,107|a^2=d,b^2=d,c^2=d,b^a=bc,c^a=cd,e^a=e^106,c^b=cd,e^b=e^106

Conjugacy classes & character table

431 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#16812053, #16812054, #16812055, …
maximal normal subgroup of	#16812053, #16812054, #16812055, …
maximal quotient of	#16812034, #16812043, C₂×Dic₄₂₈, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1712, 31);

Dic428 #13133809