C₁₃×#12404 #15698743

order	3328
nilpotency class	4
isoclinism	isoclinic class of #12316
automorphism	?
homology	Abelianization (H₁) = C₂³×C₁₃ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₁₃×#12404

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₃	#447
commutator, Frattini, cosocle	C₄×C₈	C₂³×C₁₃
prime radical	C₂×C₁₃×D₈	D₄
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃ ◁ C₄²×C₁₃ ◁ C₈×Q₈×C₁₃ ◁ G

316 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 90 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) = 1664

A minimal faithful permutation representation — 3 generators on 45 points:

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

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, 5392);