C₇₃×#10211 #32919615

order	18688
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇₃ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₇₃×#10211

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇₃	#392
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₇₃
prime radical	C₂³×C₇₃	#127
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₇₃ ◁ C₂×C₄²×C₇₃ ◁ G

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

normal subgroups only

This group has 406 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:

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

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, 3199);