C₇₃×#11914 #32921318

order	18688
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇₃ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₇₃×#11914

Subgroups

	subgroup	G/H
center, socle	C₂×C₇₃	#1530
commutator	C₄²	C₂²×C₄×C₇₃
Frattini, cosocle	#97	C₂³×C₇₃
prime radical	C₂³×C₇₃	#100
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₇₃ ◁ C₂×C₄×C₇₃ ◁ #405×C₇₃ ◁ G

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

normal subgroups only

This group has 272 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:

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

Conjugacy classes & character table

2482 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(18688, 4902);