C₁₃×#12706 #15699045

order	3328
nilpotency class	4
isoclinism	isoclinic class of #12705
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₁₃ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₁₃×#12706

Subgroups

	subgroup	G/H
center, socle	C₂×C₁₃	#1781
commutator	C₂⁴	C₂²×C₄×C₁₃
Frattini, cosocle	C₂²×D₄	C₂³×C₁₃
prime radical	C₁₃×#3485	C₂
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₃ ◁ C₂³×C₁₃ ◁ C₂²×C₁₃×#41 ◁ G

1000 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 1000 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)(16,24)(17,21)(18,30,32,22)(19,23)(20,26,34,44)(25,40)(27,39)(28,37,31,35)(29,36)(33,41,45,43)(38,42);(14,16)(15,20)(17,25)(18,26)(19,27)(21,33)(22,24)(23,34)(28,41)(29,42)(30,39)(31,43)(32,44)(35,40)(36,45)(37,38);(14,17,19,29)(15,21,23,36)(16,25,27,42)(18,31,32,28)(20,33,34,45)(22,37,30,35)(24,38,39,40)(26,43,44,41)

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, 5694);