C₁₃×#13056 #15699395

order	3328
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₁₃ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₁₃×#13056

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₃	B(4,𝔽₂)
commutator	C₂×Q₈	C₂²×C₄×C₁₃
Frattini, cosocle	C₂²×Q₈	C₂³×C₁₃
prime radical	C₂×C₁₃×2₋¹⁺⁴	C₂²
LCS	G ▷ C₂×Q₈ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃ ◁ C₂³×C₁₃ ◁ C₂²×Q₈×C₁₃ ◁ G

632 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 632 conjugacy classes of subgroups — too many to draw. Only its 118 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) = 49
transitive degree	μ_t(G) = 832

A minimal faithful permutation representation — 3 generators on 49 points:

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

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, 6044);