C₁₃×#12709 #15699048

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×#12709

Subgroups

	subgroup	G/H
center, socle	C₂×C₁₃	#1781
commutator	C₂²×C₄	C₂²×C₄×C₁₃
Frattini, cosocle	C₂²×Q₈	C₂³×C₁₃
prime radical	C₁₃×#3485	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₃ ◁ C₂³×C₁₃ ◁ C₂²×C₁₃×#41 ◁ G

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

normal subgroups only

This group has 986 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)(14,15,18,22,19,23,34,41)(16,24,27,42,26,38,32,44)(17,28,31,40,30,21,33,39)(20,35,37,29,36,43,45,25);(14,16)(15,20)(17,29)(18,32)(19,26)(21,38)(22,37)(23,36)(24,28)(25,30)(27,34)(31,35)(33,43)(39,44)(40,42)(41,45);(17,30)(21,28)(24,38)(25,29)(31,33)(35,43)(39,40)(42,44)

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, 5697);