C₇₃×#7228 #32916632

order	18688
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₄²×C₇₃ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₇₃×#7228

Subgroups

	subgroup	G/H
center	C₄×C₇₃	H(3,ℤ/4)
commutator	C₂×C₈	C₄²×C₇₃
Frattini, cosocle	C₂×#131	C₂²×C₇₃
socle	C₂×C₇₃	#1289
prime radical	C₂×#41×C₇₃	D₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₁
UCS	C₁ ◁ C₄×C₇₃ ◁ C₂×C₈×C₇₃ ◁ G

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

normal subgroups only

This group has 252 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 — 2 generators on 137 points:

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

Conjugacy classes & character table

2920 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(18688, 216);