C₅₃×#19058 #27226570

order	13568
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₅₃ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₅₃×#19058

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅₃	C₂×#120
commutator, Frattini, cosocle	C₄²	C₂⁴×C₅₃
prime radical	C₅₃×#532	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ C₂×C₄²×C₅₃ ◁ G

840 conjugacy classes of subgroups; only the 222 normal subgroups are shown.

normal subgroups only

This group has 840 conjugacy classes of subgroups — too many to draw. Only its 222 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 — 4 generators on 101 points:

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

Conjugacy classes & character table

1802 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 12046);