C₅₃×#7240 #27214752

order	13568
nilpotency class	3
isoclinism	isoclinic class of #7225
automorphism	?
homology	Abelianization (H₁) = C₄²×C₅₃ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₅₃×#7240

Subgroups

	subgroup	G/H
center	C₄×C₅₃	H(3,ℤ/4)
commutator	C₂×C₈	C₄²×C₅₃
Frattini, cosocle	#405	C₂²×C₅₃
socle	C₂×C₅₃	#1290
prime radical	C₂³×C₅₃	#108
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₁
UCS	C₁ ◁ C₄×C₅₃ ◁ C₂×C₈×C₅₃ ◁ G

148 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 148 conjugacy classes of subgroups — too many to draw. Only its 70 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) = 117
transitive degree	μ_t(G) = 3392

A minimal faithful permutation representation — 2 generators on 117 points:

(1,2,5,10,6,11,24,35,7,12,25,36,19,37,50,59)(3,13,15,38,16,39,45,61,17,40,46,56,41,62,53,64)(4,18,20,47,21,32,51,34,22,48,23,52,49,9,30,31)(8,26,27,54,28,55,57,14,29,42,33,43,58,63,60,44)(65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117);(1,3,4,14,19,41,49,54,7,17,22,44,6,16,21,43)(2,8,9,13,11,28,18,39,12,29,32,40,37,58,48,62)(5,15,20,42,50,53,30,55,25,46,23,26,24,45,51,63)(10,33,31,56,35,60,47,64,36,27,34,38,59,57,52,61)

Conjugacy classes & character table

2120 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 228);