C₅₃×#12704 #27220216

order	13568
nilpotency class	4
isoclinism	isoclinic class of #11508
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅₃ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₅₃×#12704

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅₃	C₂×#99
commutator	C₂²×C₄	C₂²×C₄×C₅₃
Frattini, cosocle	C₂²×Q₈	C₂³×C₅₃
prime radical	C₂²×#41×C₅₃	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ C₂²×C₄×C₅₃ ◁ C₂²×#41×C₅₃ ◁ G

962 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 962 conjugacy classes of subgroups — too many to draw. Only its 150 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) = 3392

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

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

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, 5692);