C₁₃×#12312 #15698651

order	3328
nilpotency class	4
isoclinism	isoclinic class of #12310
automorphism	?
homology	Abelianization (H₁) = C₂³×C₁₃ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₁₃×#12312

Subgroups

	subgroup	G/H
center, socle	C₂×C₁₃	#1619
commutator, Frattini, cosocle	C₄×C₈	C₂³×C₁₃
prime radical	C₁₃×#3292	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₃ ◁ C₂×C₄×C₁₃ ◁ C₄×C₈×C₁₃ ◁ G

372 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 372 conjugacy classes of subgroups — too many to draw. Only its 78 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)(14,15,19,23)(16,24,27,43)(17,29,33,45)(18,34,32,22)(20,35,37,26)(21,39,41,30)(25,40,28,42)(31,36,44,38);(17,30)(18,32)(21,29)(22,34)(24,40)(25,28)(26,31)(33,39)(35,44)(36,38)(41,45)(42,43);(14,16)(15,20)(17,31)(18,25)(19,27)(21,40)(22,36)(23,37)(24,29)(26,30)(28,32)(33,44)(34,38)(35,39)(41,42)(43,45)

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, 5300);