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C₂×#833 #5216

order	192
automorphism	Aut(G) order 6144
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

Splittings

C₃ ⋊ C₂×#100

Subgroups

	subgroup	G/H
center	C₂²	#217
commutator	C₂²×C₃	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₃
Fitting, prime radical	C₂²×C₃×D₄	C₂
socle	C₂²×C₃	C₂²⋊C₄
cosocle	C₂×C₃×C₄	C₂³
hypercenter	C₂²×D₄	D₃
hypocenter	C₃	C₂×#100
LCS	G ▷ C₂²×C₃ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×D₄

186 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 13
transitive degree	μ_t(G) = 48

A minimal faithful permutation representation — 3 generators on 13 points:

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

Conjugacy classes & character table

42 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#67637, #67694, #69642, …
maximal normal subgroup of	#67637, #67694, #69642, …
maximal quotient of	C₂×#4525, #71539, C₂×#4530, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 775);