#4510

order	192
isoclinism	isoclinic class of D₄₈
automorphism	Aut(G) = C₂×D₃×#442 (order 768) · Inn(G) = D₂₄ · Out(G) = C₂²×C₄
homology	Abelianization (H₁) = C₂×C₄ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Splittings

Fitting	C₂×C₃×C₁₆ ⋊ C₂
Sylow	C₃ ⋊ #359

Subgroups

	subgroup	G/H
center	C₄	D₂₄
commutator	C₃×C₈	C₂×C₄
Frattini	C₂×C₈	C₂×D₃
Fitting	C₂×C₃×C₁₆	C₂
socle	C₂×C₃	#102
prime radical	#904	C₂
cosocle	C₂×C₃×C₈	C₂²
hypercenter	C₂×C₁₆	D₃
hypocenter	C₃	#359
LCS	G ▷ C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆

56 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)(4,6,7,15,19,29,34,35,9,17,21,31,8,16,20,30)(5,10,23,27,22,26,18,28,12,24,11,14,13,25,33,32);(2,3)(4,5)(6,14)(7,18)(8,22)(9,12)(10,15)(11,20)(13,19)(16,32)(17,27)(21,33)(23,34)(24,31)(25,35)(26,30)(28,29)

Polycyclic presentation:

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

Conjugacy classes & character table

54 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#69067, #69085, #69126, …
maximal normal subgroup of	#69067, #69085, #69126, …
maximal quotient of	#67504, #67505, #67527, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 69);