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C₂×#818 #5105

order	192
isoclinism	isoclinic class of #204
automorphism	Aut(G) order 12288
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

Splittings

C₃ ⋊ C₂×#107

Subgroups

	subgroup	G/H
center	C₂³	D₁₂
commutator	C₃×C₄	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₃
Fitting	C₂²×C₃×C₈	C₂
socle, prime radical	C₂³×C₃	D₄
cosocle	C₂×C₃×C₄	C₂³
hypercenter	C₂²×C₈	D₃
hypocenter	C₃	C₂×#107
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈

130 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)(10,11,15,17,13,16,12,14);(4,5)(6,7,8,9)(11,14)(12,15)(16,17);(1,2)

Conjugacy classes & character table

60 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#67722, #67732, #67806, …
maximal normal subgroup of	#67722, #67732, #67806, …
maximal quotient of	C₂×#4458, #69731, C₄×#818, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 664);