C₁₃×#17339 #15703678

order	3328
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄²×C₁₃ · Schur multiplier (H₂) = C₂³×C₄

Aliases

direct product

C₁₃×#17339

Subgroups

	subgroup	G/H
center	C₄²×C₁₃	C₂⁴
commutator	C₂²	C₂²×C₄²×C₁₃
Frattini, cosocle	C₄²	C₂⁴×C₁₃
socle	C₂²×C₁₃	C₂²×C₄²
prime radical	C₂⁴×C₁₃	C₄²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₄²×C₁₃ ◁ G

712 conjugacy classes of subgroups; only the 518 normal subgroups are shown.

normal subgroups only

This group has 712 conjugacy classes of subgroups — too many to draw. Only its 518 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) = 832

A minimal faithful permutation representation — 4 generators on 45 points:

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

Conjugacy classes & character table

988 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3328, 10327);