Finite Group Databasebeta

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C₂×C₃⋊#95 #5208

order	192
isoclinism	isoclinic class of Dic₆
automorphism	Aut(G) order 24576
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁵

Aliases

direct product	C₂×C₃⋊#95
unambiguous semidirect product	(C₂×C₃)⋊#95

Splittings

C₃ ⋊ C₂×#95

Subgroups

	subgroup	G/H
center	C₂⁴	C₂×D₃
commutator	C₂×C₃	C₂×C₄²
Frattini	C₂³	C₂²×D₃
Fitting	C₂³×C₃×C₄	C₂
socle, prime radical	C₂⁴×C₃	C₂²
cosocle	C₂³×C₃	C₂³
hypercenter	C₂³×C₄	D₃
hypocenter	C₃	C₂×#95
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄

330 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)(6,7)(8,9)(10,11,12,13);(4,5)(7,9)(14,15,16,17);(1,2)

Conjugacy classes & character table

72 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#67624, #67674, #67675, …
maximal normal subgroup of	#67624, #67674, #67675, …
maximal quotient of	C₂×#4523, C₂×#4524, #71527, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 767);