C₅₃×#13728 #27221240

order	13568
nilpotency class	5
isoclinism	isoclinic class of #13694
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅₃ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₅₃×#13728

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅₃	C₂×D₁₆
commutator, Frattini, cosocle	C₂×C₁₆	C₂³×C₅₃
prime radical	C₂×C₅₃×D₃₂	C₂
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ C₄²×C₅₃ ◁ C₄×C₈×C₅₃ ◁ C₄×C₁₆×C₅₃ ◁ G

238 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 86 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) = 121
transitive degree	μ_t(G) = 6784

A minimal faithful permutation representation — 3 generators on 121 points:

(2,4)(5,6)(7,17)(8,23)(9,13)(10,29)(11,30)(12,19)(14,18)(15,24)(16,28)(20,37)(21,50)(22,51)(25,52)(26,56)(27,57)(31,46)(32,60)(33,61)(34,42)(35,55)(36,62)(38,41)(39,45)(40,49)(43,54)(44,53)(47,58)(48,67)(59,68)(63,65)(64,66)(69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121);(5,7,18,41,10,21,43,61,28,49,62,68,55,66,25,46,9,20,42,65,27,48,26,47,11,22,44,60,24,45,8,19)(6,12,23,39,15,32,53,51,30,58,56,67,57,63,34,37,13,31,52,64,35,59,36,40,16,33,54,50,29,38,14,17);(1,2,3,4)(6,13)(12,31)(14,34)(15,35)(16,30)(17,37)(23,52)(29,57)(32,59)(33,58)(36,53)(38,63)(39,64)(40,51)(50,67)(54,56)

Conjugacy classes & character table

3710 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 6716);