C₅₃×#12891 #27220403

order	13568
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅₃ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₅₃×#12891

Subgroups

	subgroup	G/H
center	C₂×C₄×C₅₃	C₂×D₈
commutator	C₂×C₈	C₂²×C₄×C₅₃
Frattini, cosocle	C₂²×C₈	C₂³×C₅₃
socle	C₂²×C₅₃	C₂×#107
prime radical	C₂³×C₅₃	C₂×Q₁₆
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅₃ ◁ C₂×C₄²×C₅₃ ◁ C₂×C₄×C₈×C₅₃ ◁ G

282 conjugacy classes of subgroups; only the 166 normal subgroups are shown.

normal subgroups only

This group has 282 conjugacy classes of subgroups — too many to draw. Only its 166 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) = 93
transitive degree	μ_t(G) = 6784

A minimal faithful permutation representation — 3 generators on 93 points:

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

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, 5879);