Dic₂₄ #801

order	96
isoclinism	isoclinic class of D₄₈
automorphism	Aut(G) = D₃×AGL(1,ℤ/16) (order 768) · Inn(G) = D₂₄ · Out(G) = C₂²×C₄
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₂₄

Splittings

Sylow

C₃ ⋊ Q₃₂

Subgroups

	subgroup	G/H
center	C₂	D₂₄
commutator, cosocle	C₃×C₈	C₂²
Frattini	C₈	C₂×D₃
Fitting	C₃×C₁₆	C₂
socle, prime radical	C₂×C₃	D₈
hypercenter	C₁₆	D₃
hypocenter	C₃	Q₃₂
LCS	G ▷ C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 35
transitive degree	μ_t(G) = 96

A minimal faithful permutation representation — 2 generators on 35 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);(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)(34,35)

Polycyclic presentation:

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

Conjugacy classes & character table

27 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#4449, Dic₄₈, #4520, …
maximal normal subgroup of	#4449, Dic₄₈, #4520, …
maximal quotient of	#4503, #4504, C₂×Dic₂₄, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(96, 8);

Dic24 #801