Dic₇₂ #63360

order	288
isoclinism	isoclinic class of D₁₄₄
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₇₂

Splittings

Sylow

C₉ ⋊ Q₃₂

Subgroups

	subgroup	G/H
center	C₂	D₇₂
commutator, cosocle	C₈×C₉	C₂²
Frattini	C₃×C₈	C₂×D₃
Fitting	C₉×C₁₆	C₂
socle, prime radical	C₂×C₃	D₂₄
hypercenter	C₁₆	D₉
hypocenter	C₉	Q₃₂
LCS	G ▷ C₈×C₉ ▷ C₄×C₉ ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆

36 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 41
transitive degree	μ_t(G) = 288

A minimal faithful permutation representation — 2 generators on 41 points:

(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16)(17,32,31,30,29,28,27,26,25,24,23,22,21,20,19,18)(33,34,35,36,37,38,39,40,41);(1,17,9,25)(2,18,10,26)(3,19,11,27)(4,20,12,28)(5,21,13,29)(6,22,14,30)(7,23,15,31)(8,24,16,32)(34,41)(35,40)(36,39)(37,38)

Polycyclic presentation:

2^5,3^2|a^2=e,b^2=cd,c^2=de,d^2=e,f^3=g^2,b^a=bc,c^a=cd,d^a=de,f^a=f^2g,g^a=g^2

Conjugacy classes & character table

75 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10588126, Dic₁₄₄, #10588197, …
maximal normal subgroup of	#10588126, Dic₁₄₄, #10588197, …
maximal quotient of	#10588180, #10588181, C₂×Dic₇₂, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(288, 8);

Dic72 #63360