C₃×#88123 #11928759

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

supersolvable
metabelian
solvable
monomial

Aliases

direct product

C₃×#88123

Splittings

Hall	C₃×C₁₃ ⋊ C₄⋊Q₈
Sylow	C₁₃ ⋊ C₃×C₄⋊Q₈

Subgroups

	subgroup	G/H
center	C₂²×C₃	C₂²×D₁₃
commutator, cosocle	C₂²×C₁₃	C₂³×C₃
Frattini	C₂²	C₂²×C₃×D₁₃
Fitting	C₃×C₄²×C₁₃	C₂
socle, prime radical	C₂²×C₃×C₁₃	C₂³
hypercenter	C₃×C₄²	D₁₃
hypocenter	C₁₃	C₃×C₄⋊Q₈
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×C₄²

136 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 1248

A minimal faithful permutation representation — 3 generators on 28 points:

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

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, 871);

C3×#88123 #11928759