C₁₃×#12676 #15699015

order	3328
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂×C₄²×C₁₃ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₁₃×#12676

Subgroups

	subgroup	G/H
center	C₂×C₄×C₁₃	C₂×D₈
commutator	C₈	C₂×C₄²×C₁₃
Frattini, cosocle	C₂²×C₈	C₂³×C₁₃
socle	C₂²×C₁₃	C₂×#107
prime radical	C₂³×C₁₃	C₂×Q₁₆
LCS	G ▷ 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)(14,15,17,16)(18,19,21,26,34,41,48,49,23,25,36,40,22,27,35,43)(20,28,29,38,30,45,46,37,32,44,47,39,31,42,33,24);(18,20,22,30,23,32,34,31)(19,24,27,38,25,37,41,39)(21,33,35,29,36,46,48,47)(26,42,43,28,40,45,49,44);(14,16,17,15)(19,25)(24,37)(26,40)(27,41)(28,44)(38,39)(42,45)(43,49)

Conjugacy classes & character table

832 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3328, 5664);