C₇₃×#32816 #32942220

order	18688
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₇₃ · Schur multiplier (H₂) = C₄³

Aliases

direct product

C₇₃×#32816

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³×C₇₃	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₇₃
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₇₃ ◁ #376×C₇₃ ◁ G

662 conjugacy classes of subgroups; only the 206 normal subgroups are shown.

normal subgroups only

This group has 662 conjugacy classes of subgroups — too many to draw. Only its 206 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) = 18688

A minimal faithful permutation representation — 4 generators on 121 points:

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

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, 25804);