C₁₃×#12874 #15699213

order	3328
nilpotency class	4
isoclinism	isoclinic class of #12873
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₁₃ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₁₃×#12874

Subgroups

	subgroup	G/H
center, socle	C₂×C₁₃	#1823
commutator	C₂×C₈	C₂²×C₄×C₁₃
Frattini, cosocle	C₂²×C₈	C₂³×C₁₃
prime radical	C₂×C₁₃×#41	D₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₃ ◁ C₂²×C₄×C₁₃ ◁ C₂×C₁₃×#131 ◁ G

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

normal subgroups only

This group has 376 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)(14,15,18,25,20,23,19,35)(16,26,28,39,29,21,34,38)(17,30,32,40,33,22,31,24)(27,36,43,42,44,41,45,37);(14,16)(15,21)(17,27)(18,34)(19,28)(20,29)(22,36)(23,26)(24,37)(25,39)(30,41)(31,43)(32,45)(33,44)(35,38)(40,42);(15,22,23,30)(17,31)(18,19)(21,36,26,41)(24,25,40,35)(27,43)(28,34)(32,33)(37,39,42,38)(44,45)

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, 5862);