C₁₃×#13684 #15700023

order	3328
nilpotency class	5
isoclinism	isoclinic class of #13681
automorphism	?
homology	Abelianization (H₁) = C₂³×C₁₃ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₁₃×#13684

Subgroups

	subgroup	G/H
center, socle	C₂×C₁₃	#2196
commutator, Frattini, cosocle	C₄⋊Q₈	C₂³×C₁₃
prime radical	C₁₃×#3293	C₂
LCS	G ▷ C₄⋊Q₈ ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₃ ◁ C₂²×C₁₃ ◁ C₂×C₄×C₁₃ ◁ C₁₃×C₄⋊Q₈ ◁ G

378 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 378 conjugacy classes of subgroups — too many to draw. Only its 58 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)(14,15,21,20)(16,25,29,39)(17,30,32,34)(18,19,35,24)(22,33,28,27)(23,36,37,31)(26,38,42,44)(40,43,45,41);(14,16,21,29)(15,22,20,28)(17,26,32,42)(18,33,35,27)(19,25,24,39)(23,38,37,44)(30,43,34,41)(31,45,36,40);(14,17)(15,23)(16,26)(18,34)(19,36)(20,37)(21,32)(22,38)(24,31)(25,40)(27,43)(28,44)(29,42)(30,35)(33,41)(39,45)

Conjugacy classes & character table

325 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3328, 6672);