C₁₃×#12862 #15699201

order	3328
nilpotency class	4
isoclinism	isoclinic class of #11508
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₁₃ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₁₃×#12862

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₃	C₂×#99
commutator	C₂²×C₄	C₂²×C₄×C₁₃
Frattini, cosocle	C₂²×Q₈	C₂³×C₁₃
prime radical	C₂⁴×C₁₃	C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃ ◁ C₂⁴×C₁₃ ◁ C₂³×Q₈×C₁₃ ◁ G

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

normal subgroups only

This group has 566 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) = 53
transitive degree	μ_t(G) = 832

A minimal faithful permutation representation — 3 generators on 53 points:

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

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, 5850);