C₁₃×#7239 #15693578

order	3328
nilpotency class	3
isoclinism	isoclinic class of #7225
automorphism	?
homology	Abelianization (H₁) = C₄²×C₁₃ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₁₃×#7239

Subgroups

	subgroup	G/H
center	C₄×C₁₃	H(3,ℤ/4)
commutator	C₂×C₈	C₄²×C₁₃
Frattini, cosocle	#405	C₂²×C₁₃
socle	C₂×C₁₃	#1290
prime radical	C₂³×C₁₃	#108
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₁
UCS	C₁ ◁ C₄×C₁₃ ◁ C₂×C₈×C₁₃ ◁ G

148 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 148 conjugacy classes of subgroups — too many to draw. Only its 70 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) = 77
transitive degree	μ_t(G) = 832

A minimal faithful permutation representation — 2 generators on 77 points:

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

Conjugacy classes & character table

520 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3328, 227);