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C₂×#835 #5224

order	192
isoclinism	isoclinic class of #213
automorphism	Aut(G) = D₃×#72214271 (order 6144) · Inn(G) = #67 · Out(G) = C₂×#2403
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

Splittings

C₃ ⋊ C₂×#103

Subgroups

	subgroup	G/H
center	C₂³	#67
commutator	C₃×C₄	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₃
Fitting	C₂²×C₃×Q₈	C₂
socle, prime radical	C₂³×C₃	D₄
cosocle	C₂×C₃×C₄	C₂³
hypercenter	C₂²×Q₈	D₃
hypocenter	C₃	C₂×#103
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×Q₈

162 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 17
transitive degree	μ_t(G) = 192

A minimal faithful permutation representation — 3 generators on 17 points:

(3,4,5)(10,11,13,15)(12,17,16,14);(4,5)(6,7,8,9)(11,14)(12,16)(15,17);(1,2)

Conjugacy classes & character table

48 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#69666, #69667, #69668, …
maximal normal subgroup of	#69666, #69667, #69668, …
maximal quotient of	#71554, C₂×#4529, C₂×#4535, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 783);