#4904

order	192
isoclinism	isoclinic class of D₄₈
automorphism	Aut(G) = C₂×D₃×AGL(1,ℤ/16) (order 1536) · Inn(G) = D₂₄ · Out(G) = C₂³×C₄
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Splittings

Fitting	C₂×C₃×C₁₆ ⋊ C₂
Sylow	C₃ ⋊ #508

Subgroups

	subgroup	G/H
center	C₄	D₂₄
commutator, cosocle	C₃×C₈	C₂³
Frattini	C₈	C₂²×D₃
Fitting	C₂×C₃×C₁₆	C₂
socle	C₂×C₃	C₂×D₈
hypercenter	C₂×C₁₆	D₃
hypocenter	C₃	#508
LCS	G ▷ C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆

84 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 35 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

54 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#67580, #67585, #67598, …
maximal normal subgroup of	#67580, #67585, #67598, …
maximal quotient of	#69045, C₄×D₄₈, #69059, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 463);