C₁₃×#12910 #15699249

order	3328
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₁₃ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₁₃×#12910

Subgroups

	subgroup	G/H
center	C₂×C₄×C₁₃	C₂×D₈
commutator	C₂×C₈	C₂²×C₄×C₁₃
Frattini, cosocle	C₂²×C₈	C₂³×C₁₃
socle	C₂²×C₁₃	C₂×#107
prime radical	C₂³×C₁₃	C₂×Q₁₆
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₃ ◁ C₂×C₄²×C₁₃ ◁ C₂×C₄×C₈×C₁₃ ◁ G

306 conjugacy classes of subgroups; only the 190 normal subgroups are shown.

normal subgroups only

This group has 306 conjugacy classes of subgroups — too many to draw. Only its 190 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) = 1664

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

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

Conjugacy classes & character table

676 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3328, 5898);