C₁₃×#912 #11928888

order	1248
automorphism	?
homology	Abelianization (H₁) = C₂³×C₁₃ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₁₃×#912

Splittings

Fitting	C₃×C₁₃×D₈ ⋊ C₂
Hall	C₃×C₁₃ ⋊ #135
Sylow	C₃ ⋊ C₁₃×#135

Subgroups

	subgroup	G/H
center	C₂×C₁₃	D₃×D₄
commutator, cosocle	C₃×C₄	C₂³×C₁₃
Frattini	C₄	C₂²×D₃×C₁₃
Fitting	C₃×C₁₃×D₈	C₂
socle	C₂×C₃×C₁₃	C₂×D₄
hypercenter	C₁₃×D₈	D₃
hypocenter	C₃	C₁₃×#135
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₁₃ ◁ C₄×C₁₃ ◁ C₁₃×D₈

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:

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

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1248, 1000);

C13×#912 #11928888