C₅₃×#11496 #27219008

order	13568
nilpotency class	3
isoclinism	isoclinic class of #327
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈×C₅₃ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₅₃×#11496

Subgroups

	subgroup	G/H
center	C₂×C₈×C₅₃	C₂²⋊C₄
commutator	C₂×C₄	C₂²×C₈×C₅₃
Frattini, cosocle	C₂²×C₈	C₂³×C₅₃
socle	C₂²×C₅₃	C₂×C₂²⋊C₈
prime radical	C₂³×C₅₃	C₂×M₁₆
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₅₃ ◁ C₂×C₄×C₈×C₅₃ ◁ G

296 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 296 conjugacy classes of subgroups — too many to draw. Only its 138 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) = 101
transitive degree	μ_t(G) = 6784

A minimal faithful permutation representation — 3 generators on 101 points:

(1,2)(3,9,11,6)(4,7)(5,8)(10,15,16,13)(12,14)(17,18,22,28,24,27,20,26,23,29,36,43,21,37,35,44)(19,30,33,45,32,25,31,40,34,42,46,39,38,41,47,48)(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101);(1,3,5,11)(2,6,8,9)(4,10,12,16)(7,13,14,15)(17,19,23,34)(18,25,29,41)(20,31,35,47)(21,38,24,32)(22,33,36,46)(26,39,44,45)(27,42,37,30)(28,40,43,48);(1,4,5,12)(2,7,8,14)(3,10,11,16)(6,13,9,15)(17,20,21,22,23,35,24,36)(18,26,37,28,29,44,27,43)(19,31,38,33,34,47,32,46)(25,39,30,40,41,45,42,48)

Conjugacy classes & character table

4028 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 4484);