C₁₃×#12710 #15699049

order	3328
nilpotency class	4
isoclinism	isoclinic class of #12707
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₁₃ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₁₃×#12710

Subgroups

	subgroup	G/H
center, socle	C₂×C₁₃	#1781
commutator	C₂²×C₄	C₂²×C₄×C₁₃
Frattini, cosocle	C₂²×Q₈	C₂³×C₁₃
prime radical	C₁₃×#3486	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₃ ◁ C₂³×C₁₃ ◁ C₂³×Q₈×C₁₃ ◁ G

952 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 952 conjugacy classes of subgroups — too many to draw. Only its 146 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)(14,15,19,24,20,25,17,22)(16,26,27,44,29,41,35,43)(18,31,30,42,34,23,33,40)(21,36,39,32,38,45,37,28);(14,16)(15,21)(17,27)(18,32)(19,35)(20,29)(22,37)(23,41)(24,39)(25,38)(26,31)(28,34)(30,36)(33,45)(40,43)(42,44);(14,17,20,19)(15,22,25,24)(16,27,29,35)(18,33,34,30)(21,37,38,39)(23,42,31,40)(26,43,41,44)(28,36,32,45)

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, 5698);