C₇₃×#7378 #32916782

order	18688
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂×C₈×C₇₃ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₇₃×#7378

Subgroups

	subgroup	G/H
center	C₄×C₇₃	AΣL(1,𝔽₁₆)
commutator	C₂×Q₈	C₂×C₈×C₇₃
Frattini, cosocle	#408	C₂²×C₇₃
socle	C₂×C₇₃	#1326
prime radical	C₂×#41×C₇₃	C₂×C₄
LCS	G ▷ C₂×Q₈ ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₇₃ ◁ C₂×C₄×C₇₃ ◁ C₂²×C₄×C₇₃ ◁ #408×C₇₃ ◁ G

170 conjugacy classes of subgroups; only the 46 normal subgroups are shown.

normal subgroups only

This group has 170 conjugacy classes of subgroups — too many to draw. Only its 46 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:

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

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, 366);