Dic₃₃₂ #13050581

order	1328
automorphism	Aut(G) order 217792
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₃₃₂

Splittings

Sylow

C₈₃ ⋊ Q₁₆

Subgroups

	subgroup	G/H
center	C₂	D₃₃₂
commutator, cosocle	C₄×C₈₃	C₂²
Frattini	C₄	C₂×D₈₃
Fitting	C₈×C₈₃	C₂
socle, prime radical	C₂×C₈₃	D₄
hypercenter	C₈	D₈₃
hypocenter	C₈₃	Q₁₆
LCS	G ▷ C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

18 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 99
transitive degree	μ_t(G) = 1328

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

Polycyclic presentation:

2^4,83|a^2=d,b^2=d,c^2=d,b^a=bc,c^a=cd,e^a=e^82,c^b=cd,e^b=e^82

Conjugacy classes & character table

335 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#14581736, #14581737, #14581738, …
maximal normal subgroup of	#14581736, #14581737, #14581738, …
maximal quotient of	#14581717, #14581726, C₂×Dic₃₃₂, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1328, 31);

Dic332 #13050581