C₇₃×#60402 #32969806

order	18688
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₇₃ · Schur multiplier (H₂) = C₂¹⁰

Aliases

direct product

C₇₃×#60402

Subgroups

	subgroup	G/H
center, socle	C₂×C₇₃	C₂²×#120
commutator, Frattini, cosocle	C₂³	C₂⁵×C₇₃
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₇₃ ◁ C₂²×Q₈×C₇₃ ◁ G

4778 conjugacy classes of subgroups; only the 1114 normal subgroups are shown.

normal subgroups only

This group has 4778 conjugacy classes of subgroups — too many to draw. Only its 1114 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) = 105
transitive degree	μ_t(G) = 2336

A minimal faithful permutation representation — 5 generators on 105 points:

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

Conjugacy classes & character table

4234 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(18688, 53390);