C₅₃×#8225 #27215737

order	13568
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂×C₄²×C₅₃ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₅₃×#8225

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³×C₅₃	C₂×C₂²⋊C₄
commutator	C₂×C₄	C₂×C₄²×C₅₃
Frattini, cosocle	C₂×C₄²	C₂³×C₅₃
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₅₃ ◁ C₄³×C₅₃ ◁ G

444 conjugacy classes of subgroups; only the 178 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 178 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) = 89
transitive degree	μ_t(G) = 13568

A minimal faithful permutation representation — 3 generators on 89 points:

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

Conjugacy classes & character table

2756 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 1213);