C₁₃×#1010 #11929297

order	1248
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₁₃ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₁₃×#1010

Splittings

Fitting	C₃×C₁₃×#41 ⋊ C₂
Hall	C₃×C₁₃ ⋊ 2₋¹⁺⁴
Sylow	C₃ ⋊ C₁₃×2₋¹⁺⁴

Subgroups

	subgroup	G/H
center	C₂×C₁₃	C₂³×D₃
commutator, cosocle	C₂×C₃	C₂⁴×C₁₃
Frattini	C₂	C₂³×D₃×C₁₃
Fitting	C₃×C₁₃×#41	C₂
socle	C₂×C₃×C₁₃	C₂⁴
hypercenter	C₁₃×#41	D₃
hypocenter	C₃	C₁₃×2₋¹⁺⁴
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₁₃ ◁ C₁₃×#41

292 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 624

A minimal faithful permutation representation — 4 generators on 32 points:

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

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1248, 1409);

C13×#1010 #11929297