C₁₃×#12267 #15698606

order	3328
nilpotency class	4
isoclinism	isoclinic class of #12148
automorphism	?
homology	Abelianization (H₁) = C₂³×C₁₃ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₁₃×#12267

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₃	B(4,𝔽₂)
commutator, Frattini, cosocle	C₂×C₂²⋊C₄	C₂³×C₁₃
prime radical	C₁₃×#392	C₂²
LCS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃ ◁ C₂³×C₁₃ ◁ C₂×C₁₃×C₂²⋊C₄ ◁ G

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

normal subgroups only

This group has 414 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) = 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)(14,15,18,23)(16,25,28,42)(17,20,29,22)(19,24,32,37)(21,35,39,26)(27,36,45,41)(30,33,43,38)(31,34,44,40);(15,20)(16,26)(17,29)(21,36)(22,23)(24,34)(25,33)(27,43)(28,35)(30,45)(31,44)(37,40)(38,42)(39,41);(14,16,18,28)(15,21,37,42)(17,30,29,43)(19,26,32,35)(20,33,40,41)(22,38,34,36)(23,39,24,25)(27,44,45,31)

Conjugacy classes & character table

364 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3328, 5255);