C₇₃×#13352 #32922756

order	18688
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇₃ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₇₃×#13352

Subgroups

	subgroup	G/H
center, socle	C₂×C₇₃	#2021
commutator, Frattini, cosocle	C₂×#41	C₂³×C₇₃
prime radical	#585×C₇₃	C₂²
LCS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₇₃ ◁ C₂×C₄×C₇₃ ◁ C₂×#41×C₇₃ ◁ G

682 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 682 conjugacy classes of subgroups — too many to draw. Only its 94 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) = 137
transitive degree	μ_t(G) = 4672

A minimal faithful permutation representation — 3 generators on 137 points:

(1,2,7,12)(3,13,17,39)(4,18,21,36)(5,10,23,35)(6,24,26,11)(8,27,30,15)(9,25,33,20)(14,40,43,64)(16,46,48,38)(19,50,52,63)(22,53,55,34)(28,58,31,61)(29,45,56,57)(32,54,62,51)(37,47,59,42)(41,49,44,60)(65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137);(1,3)(2,8)(4,14)(5,15)(6,16)(7,17)(9,31)(10,13)(11,29)(12,30)(18,49)(19,41)(20,42)(21,43)(22,45)(23,27)(24,56)(25,47)(26,48)(28,33)(32,37)(34,38)(35,39)(36,60)(40,50)(44,52)(46,53)(51,58)(54,61)(55,57)(59,62)(63,64);(1,4,7,21)(2,9,12,33)(3,14,17,43)(5,19,23,52)(6,25,26,20)(8,28,30,31)(10,32,35,62)(11,36,24,18)(13,37,39,59)(15,44,27,41)(16,47,48,42)(22,54,55,51)(29,49,56,60)(34,63,53,50)(38,64,46,40)(45,58,57,61)

Conjugacy classes & character table

2044 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(18688, 6340);