#4515

order	192
isoclinism	isoclinic class of #213
automorphism	Aut(G) = C₂×C₄×D₃×D₄ (order 384) · Inn(G) = #67 · Out(G) = C₂²×C₄
homology	Abelianization (H₁) = C₂×C₈ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Splittings

Fitting	C₃×M₃₂ ⋊ C₂
Sylow	C₃ ⋊ #350

Subgroups

	subgroup	G/H
center	C₈	#67
commutator	C₃×C₄	C₂×C₈
Frattini	C₂×C₈	C₂×D₃
Fitting	C₃×M₃₂	C₂
socle	C₂×C₃	C₂²⋊C₈
prime radical	#235	C₄
cosocle	C₂×C₃×C₈	C₂²
hypercenter	M₃₂	D₃
hypocenter	C₃	#350
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ M₃₂

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

Polycyclic presentation:

2^6,3|b^2=c,c^2=e,d^2=f,e^2=f,b^a=bd,d^a=df,g^a=g^2,d^b=df

Conjugacy classes & character table

48 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#350, #69125, #69258, …
maximal normal subgroup of	D₃×#350, #69125, #69258, …
maximal quotient of	#67515, #67518, #67523, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 74);