C₂×C₁₃×#1425 #15699974

order	3328
nilpotency class	5
isoclinism	isoclinic class of D₃₂
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₁₃ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₂×C₁₃×#1425

Subgroups

	subgroup	G/H
center	C₂×C₄×C₁₃	D₁₆
commutator	C₁₆	C₂²×C₄×C₁₃
Frattini, cosocle	C₂×C₁₆	C₂³×C₁₃
socle	C₂²×C₁₃	#366
prime radical	C₂³×C₁₃	Q₃₂
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₃ ◁ C₂²×C₄×C₁₃ ◁ C₂²×C₈×C₁₃ ◁ C₂²×C₁₃×C₁₆ ◁ G

182 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 182 conjugacy classes of subgroups — too many to draw. Only its 126 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) = 79
transitive degree	μ_t(G) = 1664

A minimal faithful permutation representation — 3 generators on 79 points:

(1,2)(67,68,69,70,71,72,73,74,75,76,77,78,79);(3,4,7,13,10,15,9,31)(5,16,19,41,22,43,21,53)(6,23,25,54,27,56,26,57)(8,29,28,39,30,58,24,14)(11,32,33,49,35,48,59,18)(12,36,37,46,38,45,55,17)(20,51,50,63,52,66,47,42)(34,44,40,60,61,65,62,64);(3,5,17,44,8,20,25,32,7,19,36,60,28,50,27,49,10,22,46,65,30,52,26,48,9,21,45,64,24,47,6,18)(4,11,23,42,14,34,55,53,31,59,57,66,58,62,38,43,15,35,56,63,39,61,37,41,13,33,54,51,29,40,12,16)

Conjugacy classes & character table

988 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3328, 6623);