Dic₆₈ #63221

order	272
isoclinism	isoclinic class of #63219
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₆₈

Splittings

Sylow

C₁₇ ⋊ Q₁₆

Subgroups

	subgroup	G/H
center	C₂	D₆₈
commutator, cosocle	C₄×C₁₇	C₂²
Frattini	C₄	C₂×D₁₇
Fitting	C₈×C₁₇	C₂
socle, prime radical	C₂×C₁₇	D₄
hypercenter	C₈	D₁₇
hypocenter	C₁₇	Q₁₆
LCS	G ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

18 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 33
transitive degree	μ_t(G) = 272

A minimal faithful permutation representation — 2 generators on 33 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);(1,9,5,13)(2,10,6,14)(3,11,7,15)(4,12,8,16)(18,33)(19,32)(20,31)(21,30)(22,29)(23,28)(24,27)(25,26)

Polycyclic presentation:

2^4,17|a^2=d,b^2=cd,c^2=d,b^a=bc,c^a=cd,e^a=e^16

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10587471, Dic₁₃₆, #10587498, …
maximal normal subgroup of	#10587471, Dic₁₃₆, #10587498, …
maximal quotient of	#10587487, #10587489, C₂×Dic₆₈, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(272, 8);

Dic68 #63221