C₁₃×#12775 #15699114

order	3328
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₁₃ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₁₃×#12775

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₃	B(4,𝔽₂)
commutator	C₂×Q₈	C₂²×C₄×C₁₃
Frattini, cosocle	C₂²×Q₈	C₂³×C₁₃
prime radical	C₂²×D₄×C₁₃	D₄
LCS	G ▷ C₂×Q₈ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃ ◁ C₂³×C₁₃ ◁ C₂²×Q₈×C₁₃ ◁ G

740 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 740 conjugacy classes of subgroups — too many to draw. Only its 142 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) = 49
transitive degree	μ_t(G) = 832

A minimal faithful permutation representation — 3 generators on 49 points:

(1,2,3,4,5,6,7,8,9,10,11,12,13)(14,15)(16,17)(18,19,23,26,24,29,22,28)(20,30,33,47,34,45,32,48)(21,35,38,44,39,27,37,46)(25,40,41,36,43,49,42,31);(18,20,24,34)(19,25,29,43)(21,36,39,31)(22,33,23,32)(26,41,28,42)(27,45,35,30)(37,49,38,40)(44,47,46,48);(15,17)(19,26)(22,23)(25,41)(27,46)(28,29)(30,47)(32,33)(35,44)(37,38)(40,49)(42,43)(45,48)

Conjugacy classes & character table

442 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3328, 5763);