C₁₃×#7427 #15693766

order	3328
nilpotency class	5
isoclinism	isoclinic class of #7426
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₁₃ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₁₃×#7427

Subgroups

	subgroup	G/H
center, socle	C₂×C₁₃	#1351
commutator	C₂×C₂²⋊C₄	C₂×C₄×C₁₃
Frattini, cosocle	#381	C₂²×C₁₃
prime radical	C₂⁴×C₁₃	C₂²⋊C₄
LCS	G ▷ C₂×C₂²⋊C₄ ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₃ ◁ C₂³×C₁₃ ◁ C₂⁴×C₁₃ ◁ C₁₃×#381 ◁ G

214 conjugacy classes of subgroups; only the 42 normal subgroups are shown.

normal subgroups only

This group has 214 conjugacy classes of subgroups — too many to draw. Only its 42 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 — 2 generators on 45 points:

(1,2,3,4,5,6,7,8,9,10,11,12,13)(14,15,18,22,19,23,32,39)(16,24,26,40,27,41,43,45)(17,28,30,44,31,21,36,37)(20,33,34,25,35,42,38,29);(14,16)(15,20)(17,29,31,25)(18,26)(19,27)(21,24,28,41)(22,38)(23,35)(30,42,36,33)(32,43)(34,39)(37,40,44,45)

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, 415);