C₇₃×#7851 #32917255

order	18688
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂×C₄²×C₇₃ · Schur multiplier (H₂) = C₂²×C₄

Aliases

direct product

C₇₃×#7851

Subgroups

	subgroup	G/H
center	C₂×C₄²×C₇₃	C₂³
commutator	C₂³	C₂×C₄²×C₇₃
Frattini, cosocle	C₂×C₄²	C₂³×C₇₃
socle	C₂³×C₇₃	C₂×C₄²
prime radical	C₂⁴×C₇₃	C₄²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂×C₄²×C₇₃ ◁ G

610 conjugacy classes of subgroups; only the 286 normal subgroups are shown.

normal subgroups only

This group has 610 conjugacy classes of subgroups — too many to draw. Only its 286 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) = 9344

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

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

Conjugacy classes & character table

6424 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(18688, 839);