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Dic₁₂₆ #92608

order	504
isoclinism	isoclinic class of Dic₁₂₆
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

Dic₁₂₆

Splittings

Hall	C₇×C₉ ⋊ Q₈
Sylow	C₉ ⋊ Dic₁₄
Sylow	C₇ ⋊ Dic₁₈

Subgroups

	subgroup	G/H
center	C₂	C₂×D₆₃
commutator, cosocle	C₂×C₇×C₉	C₂²
Frattini	C₂×C₃	C₂×D₂₁
Fitting	C₄×C₇×C₉	C₂
socle, prime radical	C₂×C₃×C₇	C₂×D₃
hypercenter	C₄	D₆₃
hypocenter	C₇×C₉	Q₈
LCS	G ▷ C₂×C₇×C₉ ▷ C₇×C₉
UCS	C₁ ◁ C₂ ◁ C₄

36 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 504

A minimal faithful permutation representation — 2 generators on 24 points:

(1,2,3,4)(5,8,7,6)(9,10,11,12,13,14,15,16,17)(18,19,20,21,22,23,24);(1,5,3,7)(2,6,4,8)(10,17)(11,16)(12,15)(13,14)(19,24)(20,23)(21,22)

Polycyclic presentation:

2^3,3,7,3|a^2=c,b^2=c,d^3=f^2,b^a=bc,d^a=d^2f,e^a=e^6,f^a=f^2

Conjugacy classes & character table

129 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#11758856, #11758859, #11758865, …
maximal normal subgroup of	#11758856, #11758859, #11758865, …
maximal quotient of	#11758927, C₄⋊Dic₆₃, C₂×Dic₁₂₆, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(504, 34);