C₃×#88172 #11928808

order	1248
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₃×#88172

Splittings

Fitting	C₃×C₁₃×Q₁₆ ⋊ C₂
Hall	C₃×C₁₃ ⋊ #137
Sylow	C₁₃ ⋊ C₃×#137

Subgroups

	subgroup	G/H
center	C₂×C₃	D₄×D₁₃
commutator, cosocle	C₄×C₁₃	C₂³×C₃
Frattini	C₄	C₂²×C₃×D₁₃
Fitting	C₃×C₁₃×Q₁₆	C₂
socle	C₂×C₃×C₁₃	C₂×D₄
prime radical	C₃×#6131	C₂
hypercenter	C₃×Q₁₆	D₁₃
hypocenter	C₁₃	C₃×#137
LCS	G ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₃ ◁ C₃×C₄ ◁ C₃×Q₁₆

120 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 120 conjugacy classes of subgroups — too many to draw. Only its 54 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)(17,19,21,25)(18,22,24,31)(20,29,30,27)(23,32,28,26);(17,18,21,24)(19,26,25,32)(20,28,30,23)(22,29,31,27);(5,6)(7,8)(9,10)(11,12)(13,14)(15,16)(19,25)(22,31)(26,32)(27,29)

Conjugacy classes & character table

159 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1248, 920);