C₇₃×#13612 #32923016

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

Aliases

direct product

C₇₃×#13612

Subgroups

	subgroup	G/H
center, socle	C₂×C₇₃	#2127
commutator	C₄²	C₂²×C₄×C₇₃
Frattini, cosocle	C₄⋊Q₈	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

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

normal subgroups only

This group has 304 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:

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

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, 6600);