Dic₂₃₆ #11745662

order	944
automorphism	Aut(G) order 109504
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₂₃₆

Splittings

Sylow

C₅₉ ⋊ Q₁₆

Subgroups

	subgroup	G/H
center	C₂	D₂₃₆
commutator, cosocle	C₄×C₅₉	C₂²
Frattini	C₄	C₂×D₅₉
Fitting	C₈×C₅₉	C₂
socle, prime radical	C₂×C₅₉	D₄
hypercenter	C₈	D₅₉
hypocenter	C₅₉	Q₁₆
LCS	G ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

18 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 75
transitive degree	μ_t(G) = 944

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

Polycyclic presentation:

2^4,59|a^2=d,b^2=cd,c^2=d,b^a=bc,c^a=cd,e^a=e^58

Conjugacy classes & character table

239 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#14273671, #14273672, #14273673, …
maximal normal subgroup of	#14273671, #14273672, #14273673, …
maximal quotient of	#14273652, #14273661, C₂×Dic₂₃₆, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(944, 7);

Dic236 #11745662