C₅₃×#11469 #27218981

order	13568
nilpotency class	3
isoclinism	isoclinic class of #1510
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈×C₅₃ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₅₃×#11469

Subgroups

	subgroup	G/H
center	C₂×C₄×C₅₃	C₂×C₂²⋊C₄
commutator	C₂×C₄	C₂²×C₈×C₅₃
Frattini, cosocle	C₂²×C₈	C₂³×C₅₃
socle	C₂²×C₅₃	C₂×C₂²⋊C₈
prime radical	C₅₃×#525	C₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅₃ ◁ C₂³×C₈×C₅₃ ◁ G

386 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 386 conjugacy classes of subgroups — too many to draw. Only its 130 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) = 3392

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

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

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, 4457);