Finite Group Databasebeta

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C₁₃×#805 #11928076

order	1248
isoclinism	isoclinic class of #204
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₁₃ · Schur multiplier (H₂) = C₂

Aliases

Splittings

Fitting	C₃×C₄²×C₁₃ ⋊ C₂
Hall	C₃×C₁₃ ⋊ #104
Sylow	C₃ ⋊ C₁₃×#104

Subgroups

	subgroup	G/H
center	C₄×C₁₃	D₁₂
commutator	C₃×C₄	C₂×C₄×C₁₃
Frattini	C₂×C₄	C₂×D₃×C₁₃
Fitting	C₃×C₄²×C₁₃	C₂
socle	C₂×C₃×C₁₃	C₂²⋊C₄
prime radical	C₁₃×#235	C₂
cosocle	C₂×C₃×C₄	C₂²×C₁₃
hypercenter	C₄²×C₁₃	D₃
hypocenter	C₃	C₁₃×#104
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₄×C₁₃ ◁ C₂×C₄×C₁₃ ◁ C₄²×C₁₃

88 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 312

A minimal faithful permutation representation — 2 generators on 24 points:

(1,2,3,4,5,6,7,8,9,10,11,12,13)(14,15,16)(18,21,22,23);(15,16)(17,18)(19,21)(20,22)(23,24)

Conjugacy classes & character table

390 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1248, 188);