C₇₃×#13608 #32923012

order	18688
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇₃ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₇₃×#13608

Subgroups

	subgroup	G/H
center, socle	C₂×C₇₃	#2127
commutator	C₄²	C₂²×C₄×C₇₃
Frattini, cosocle	#124	C₂³×C₇₃
prime radical	C₂×#41×C₇₃	D₄
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₇₃ ◁ C₂²×C₇₃ ◁ C₂²×C₄×C₇₃ ◁ #536×C₇₃ ◁ G

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

normal subgroups only

This group has 308 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) = 137
transitive degree	μ_t(G) = 4672

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

(1,2)(3,13,16,40)(4,9)(5,21,23,55)(6,25,27,11)(7,12)(8,15,29,46)(10,24)(14,39,43,61)(17,36,26,41)(18,49,22,54)(19,51,53,34)(20,35)(28,45,59,64)(30,47,31,48)(32,42,60,63)(33,50)(37,56)(38,57)(44,58,52,62)(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);(1,3,7,17)(2,8,12,31)(4,14,20,44)(5,15,23,48)(6,26,27,16)(9,32,35,28)(10,36,38,13)(11,29,25,30)(18,42,22,45)(19,52,53,43)(21,41,55,40)(24,46,57,47)(33,58,37,39)(34,60,51,59)(49,62,54,61)(50,63,56,64);(1,4,7,20)(2,9,12,35)(3,14,17,44)(5,22,23,18)(6,19,27,53)(8,28,31,32)(10,37,38,33)(11,34,25,51)(13,39,36,58)(15,45,48,42)(16,43,26,52)(21,54,55,49)(24,56,57,50)(29,59,30,60)(40,61,41,62)(46,64,47,63)

Conjugacy classes & character table

1825 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(18688, 6596);