C₇₃×#7401 #32916805

order	18688
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₇₃ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₇₃×#7401

Subgroups

	subgroup	G/H
center, socle	C₂×C₇₃	#1343
commutator	C₂²×C₈	C₂×C₄×C₇₃
Frattini, cosocle	#408	C₂²×C₇₃
prime radical	C₂×#41×C₇₃	D₄
LCS	G ▷ C₂²×C₈ ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₇₃ ◁ C₂×C₄×C₇₃ ◁ C₂²×C₄×C₇₃ ◁ #408×C₇₃ ◁ G

234 conjugacy classes of subgroups; only the 42 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 42 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 — 2 generators on 137 points:

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

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, 389);