C₇₃×#12851 #32922255

order	18688
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈×C₇₃ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₇₃×#12851

Subgroups

	subgroup	G/H
center	C₄×C₇₃	#381
commutator	C₂³	C₂²×C₈×C₇₃
Frattini, cosocle	C₂²×C₈	C₂³×C₇₃
socle	C₂×C₇₃	#1817
prime radical	C₂⁴×C₇₃	C₂×C₈
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₇₃ ◁ C₂²×C₈×C₇₃ ◁ G

416 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 416 conjugacy classes of subgroups — too many to draw. Only its 162 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 — 3 generators on 105 points:

(1,2,4,9,5,10,16,23,6,7,15,18,17,19,28,29)(3,11,13,24,14,25,27,32,12,8,20,21,26,22,30,31)(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);(2,7)(4,15)(8,11)(10,19)(13,20)(16,28)(22,25)(27,30);(1,3)(2,8,7,11)(4,13)(5,14)(6,12)(9,21,18,24)(10,22,19,25)(15,20)(16,27)(17,26)(23,31,29,32)(28,30)

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, 5839);