C₇₃×#9004 #32918408

order	18688
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇₃ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₇₃×#9004

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³×C₇₃	#120
commutator	C₄²	C₂²×C₄×C₇₃
Frattini, cosocle	C₂×C₄²	C₂³×C₇₃
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₇₃ ◁ C₂×C₄²×C₇₃ ◁ G

458 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 134 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) = 109
transitive degree	μ_t(G) = 18688

A minimal faithful permutation representation — 3 generators on 109 points:

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

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, 1992);