C₇₃×#10225 #32919629

order	18688
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇₃ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₇₃×#10225

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇₃	#392
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₇₃
prime radical	C₇₃×#3150	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₇₃ ◁ C₂×C₄²×C₇₃ ◁ G

574 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 574 conjugacy classes of subgroups — too many to draw. Only its 122 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) = 113
transitive degree	μ_t(G) = 4672

A minimal faithful permutation representation — 3 generators on 113 points:

(1,2)(3,6)(4,7)(5,8)(9,10,15,20)(11,21,24,32)(12,17,28,35)(13,19,14,18)(16,25,31,23)(22,37,38,33)(26,40,39,30)(27,36,29,34)(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113);(2,6)(3,4)(7,8)(9,11)(10,16)(12,26)(13,25)(14,23)(15,24)(17,33)(18,32)(19,21)(20,31)(22,27)(28,39)(29,38)(30,34)(35,37)(36,40);(2,7)(3,4)(6,8)(9,12)(10,17)(11,22)(13,27)(14,29)(15,28)(16,30)(18,34)(19,36)(20,35)(21,33)(23,26)(24,38)(25,39)(31,40)(32,37)

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, 3213);