Dic₄₈ #4450

order	192
isoclinism	isoclinic class of D₉₆
automorphism	Aut(G) = D₃×AGL(1,ℤ/32) (order 3072) · Inn(G) = D₄₈ · Out(G) = C₂²×C₈
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₄₈

Splittings

Sylow

C₃ ⋊ Q₆₄

Subgroups

	subgroup	G/H
center	C₂	D₄₈
commutator, cosocle	C₃×C₁₆	C₂²
Frattini	C₁₆	C₂×D₃
Fitting	C₃×C₃₂	C₂
socle, prime radical	C₂×C₃	D₁₆
hypercenter	C₃₂	D₃
hypocenter	C₃	Q₆₄
LCS	G ▷ C₃×C₁₆ ▷ C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆ ◁ C₃₂

30 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 67
transitive degree	μ_t(G) = 192

A minimal faithful permutation representation — 2 generators on 67 points:

(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32)(33,64,63,62,61,60,59,58,57,56,55,54,53,52,51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34)(65,66,67);(1,33,17,49)(2,34,18,50)(3,35,19,51)(4,36,20,52)(5,37,21,53)(6,38,22,54)(7,39,23,55)(8,40,24,56)(9,41,25,57)(10,42,26,58)(11,43,27,59)(12,44,28,60)(13,45,29,61)(14,46,30,62)(15,47,31,63)(16,48,32,64)(66,67)

Polycyclic presentation:

2^6,3|a^2=f,b^2=cd,c^2=de,d^2=ef,e^2=f,b^a=bc,c^a=cd,d^a=de,e^a=ef,g^a=g^2

Conjugacy classes & character table

51 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#67434, Dic₉₆, #67608, …
maximal normal subgroup of	#67434, Dic₉₆, #67608, …
maximal quotient of	#67591, #67597, C₂×Dic₄₈, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 9);

Dic48 #4450