C₃×#88204 #11928840

order	1248
isoclinism	isoclinic class of #6104
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₃×#88204

Splittings

Fitting	C₃×C₁₃×#41 ⋊ C₂
Hall	C₃×C₁₃ ⋊ #135
Sylow	C₁₃ ⋊ C₃×#135

Subgroups

	subgroup	G/H
center	C₃×C₄	C₂²⋊D₁₃
commutator, cosocle	C₄×C₁₃	C₂³×C₃
Frattini	C₄	C₂²×C₃×D₁₃
Fitting	C₃×C₁₃×#41	C₂
socle	C₂×C₃×C₁₃	C₂×D₄
hypercenter	C₃×#41	D₁₃
hypocenter	C₁₃	C₃×#135
LCS	G ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₃×C₄ ◁ C₂×C₃×C₄ ◁ C₃×#41

124 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 124 conjugacy classes of subgroups — too many to draw. Only its 58 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

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

Conjugacy classes & character table

222 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1248, 952);