C₁₃×#12876 #15699215

order	3328
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₁₃ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₁₃×#12876

Subgroups

	subgroup	G/H
center	C₄×C₁₃	#481
commutator	C₂×C₈	C₂²×C₄×C₁₃
Frattini, cosocle	C₂²×C₈	C₂³×C₁₃
socle	C₂×C₁₃	#1824
prime radical	C₂⁴×C₁₃	Q₁₆
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₁₃ ◁ C₂²×C₄×C₁₃ ◁ C₂²×C₁₃×M₁₆ ◁ G

370 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 138 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) = 45
transitive degree	μ_t(G) = 416

A minimal faithful permutation representation — 3 generators on 45 points:

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

Conjugacy classes & character table

520 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3328, 5864);