Dic₁₂₄ #92472

order	496
isoclinism	isoclinic class of #92470
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₂₄

Splittings

Sylow

C₃₁ ⋊ Q₁₆

Subgroups

	subgroup	G/H
center	C₂	D₁₂₄
commutator, cosocle	C₄×C₃₁	C₂²
Frattini	C₄	C₂×D₃₁
Fitting	C₈×C₃₁	C₂
socle, prime radical	C₂×C₃₁	D₄
hypercenter	C₈	D₃₁
hypocenter	C₃₁	Q₁₆
LCS	G ▷ C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

18 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 47
transitive degree	μ_t(G) = 496

A minimal faithful permutation representation — 2 generators on 47 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);(1,9,5,13)(2,10,6,14)(3,11,7,15)(4,12,8,16)(18,47)(19,46)(20,45)(21,44)(22,43)(23,42)(24,41)(25,40)(26,39)(27,38)(28,37)(29,36)(30,35)(31,34)(32,33)

Polycyclic presentation:

2^4,31|a^2=d,b^2=cd,c^2=d,b^a=bc,c^a=cd,e^a=e^30

Conjugacy classes & character table

127 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#11758388, Dic₂₄₈, #11758415, …
maximal normal subgroup of	#11758388, Dic₂₄₈, #11758415, …
maximal quotient of	#11758404, #11758406, C₂×Dic₁₂₄, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(496, 7);

Dic124 #92472