C₁₃×#12277 #15698616

order	3328
nilpotency class	4
isoclinism	isoclinic class of #12155
automorphism	?
homology	Abelianization (H₁) = C₂³×C₁₃ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₁₃×#12277

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₃	B(4,𝔽₂)
commutator, Frattini, cosocle	C₂×C₄⋊C₄	C₂³×C₁₃
prime radical	C₂³×C₁₃	#120
LCS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃ ◁ C₂³×C₁₃ ◁ C₂×C₁₃×C₄⋊C₄ ◁ G

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

normal subgroups only

This group has 318 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) = 61
transitive degree	μ_t(G) = 832

A minimal faithful permutation representation — 3 generators on 61 points:

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

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, 5265);