Dic₂₈₄ #11765856

order	1136
automorphism	Aut(G) order 159040
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₂₈₄

Splittings

Sylow

C₇₁ ⋊ Q₁₆

Subgroups

	subgroup	G/H
center	C₂	D₂₈₄
commutator, cosocle	C₄×C₇₁	C₂²
Frattini	C₄	C₂×D₇₁
Fitting	C₈×C₇₁	C₂
socle, prime radical	C₂×C₇₁	D₄
hypercenter	C₈	D₇₁
hypocenter	C₇₁	Q₁₆
LCS	G ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

18 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 87
transitive degree	μ_t(G) = 1136

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

Polycyclic presentation:

2^4,71|a^2=d,b^2=d,c^2=d,b^a=bc,c^a=cd,e^a=e^70,c^b=cd,e^b=e^70

Conjugacy classes & character table

287 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#14556072, #14556073, #14556074, …
maximal normal subgroup of	#14556072, #14556073, #14556074, …
maximal quotient of	#14556053, #14556062, C₂×Dic₂₈₄, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1136, 31);

Dic284 #11765856