C₁₃×#919 #11928895

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

Aliases

direct product

C₁₃×#919

Splittings

Fitting	C₃×C₁₃×Q₁₆ ⋊ 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×Q₁₆	C₂
socle	C₂×C₃×C₁₃	C₂×D₄
hypercenter	C₁₃×Q₁₆	D₃
hypocenter	C₃	C₁₃×#135
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₁₃ ◁ C₄×C₁₃ ◁ C₁₃×Q₁₆

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)(17,19,21,25)(18,23,22,31)(20,29,30,27)(24,32,28,26);(17,18,21,22)(19,26,25,32)(20,28,30,24)(23,29,31,27);(15,16)(18,22)(19,25)(24,28)(27,29)

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, 1007);

C13×#919 #11928895