C₁₃×#7373 #15693712

order	3328
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂×C₈×C₁₃ · Schur multiplier (H₂) = C₄

Aliases

direct product

C₁₃×#7373

Subgroups

	subgroup	G/H
center	C₄×C₁₃	#353
commutator	C₄²	C₂×C₈×C₁₃
Frattini, cosocle	#423	C₂²×C₁₃
socle	C₂×C₁₃	#1325
prime radical	C₂³×C₁₃	#101
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₁₃ ◁ C₂×C₄×C₁₃ ◁ C₂²×C₄×C₁₃ ◁ C₁₃×#423 ◁ G

152 conjugacy classes of subgroups; only the 46 normal subgroups are shown.

normal subgroups only

This group has 152 conjugacy classes of subgroups — too many to draw. Only its 46 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 — 2 generators on 45 points:

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

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, 361);