C₁₃×#951 #11928927

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

Aliases

direct product

C₁₃×#951

Splittings

Hall	C₃×C₁₃ ⋊ #137
Sylow	C₃ ⋊ C₁₃×#137

Subgroups

	subgroup	G/H
center	C₂×C₁₃	C₂×#67
commutator, cosocle	C₃×C₄	C₂³×C₁₃
Frattini	C₄	C₂²×D₃×C₁₃
Fitting, prime radical	C₃×C₁₃×#41	C₂
socle	C₂×C₃×C₁₃	C₂×D₄
hypercenter	C₁₃×#41	D₃
hypocenter	C₃	C₁₃×#137
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₁₃ ◁ C₂×C₄×C₁₃ ◁ C₁₃×#41

120 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 120 conjugacy classes of subgroups — too many to draw. Only its 58 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:

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

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, 1039);