Dic₂₉₂ #11924019

order	1168
automorphism	Aut(G) order 168192
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₂₉₂

Splittings

Sylow

C₇₃ ⋊ Q₁₆

Subgroups

	subgroup	G/H
center	C₂	D₂₉₂
commutator, cosocle	C₄×C₇₃	C₂²
Frattini	C₄	C₂×D₇₃
Fitting	C₈×C₇₃	C₂
socle, prime radical	C₂×C₇₃	D₄
hypercenter	C₈	D₇₃
hypocenter	C₇₃	Q₁₆
LCS	G ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

18 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 89
transitive degree	μ_t(G) = 1168

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

Polycyclic presentation:

2^4,73|a^2=d,b^2=d,c^2=d,b^a=bc,c^a=cd,e^a=e^72,c^b=cd,e^b=e^72

Conjugacy classes & character table

295 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#14556535, #14556536, #14556537, …
maximal normal subgroup of	#14556535, #14556536, #14556537, …
maximal quotient of	#14556516, #14556525, C₂×Dic₂₉₂, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1168, 31);

Dic292 #11924019