C₃×#88158 #11928794

order	1248
isoclinism	isoclinic class of #6095
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₃×#88158

Splittings

Fitting	C₂×C₃×C₈×C₁₃ ⋊ C₂
Hall	C₃×C₁₃ ⋊ #135
Sylow	C₁₃ ⋊ C₃×#135

Subgroups

	subgroup	G/H
center	C₃×C₄	D₅₂
commutator, cosocle	C₄×C₁₃	C₂³×C₃
Frattini	C₄	C₂²×C₃×D₁₃
Fitting	C₂×C₃×C₈×C₁₃	C₂
socle	C₂×C₃×C₁₃	C₂×D₄
hypercenter	C₂×C₃×C₈	D₁₃
hypocenter	C₁₃	C₃×#135
LCS	G ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₃×C₄ ◁ C₂×C₃×C₄ ◁ C₂×C₃×C₈

124 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 624

A minimal faithful permutation representation — 3 generators on 32 points:

(1,2,3)(4,5,7,9,11,13,15,16,14,12,10,8,6)(17,20,21,28,22,29,19,26)(18,24,23,27,25,31,30,32);(17,19,22,21)(18,23,25,30)(20,26,29,28)(24,27,31,32);(5,6)(7,8)(9,10)(11,12)(13,14)(15,16)(17,18)(19,23)(20,27)(21,30)(22,25)(24,28)(26,31)(29,32)

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1248, 906);

C3×#88158 #11928794