C₇₃×#11676 #32921080

order	18688
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇₃ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₇₃×#11676

Subgroups

	subgroup	G/H
center	C₂×C₄×C₇₃	C₂×D₈
commutator	C₂×C₈	C₂²×C₄×C₇₃
Frattini, cosocle	C₄×C₈	C₂³×C₇₃
socle	C₂²×C₇₃	C₂×#102
prime radical	C₂³×C₇₃	#102
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₇₃ ◁ C₂×C₄²×C₇₃ ◁ C₂×C₄×C₈×C₇₃ ◁ G

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

normal subgroups only

This group has 316 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) = 9344

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

(1,2,4,7,5,3,8,6)(9,10)(11,20)(12,17)(13,26)(14,30)(15,19)(16,22)(18,28)(21,34)(23,40)(24,37)(25,35)(27,33)(29,36)(31,38)(32,39)(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);(1,3)(2,5)(4,6)(7,8)(9,11,15,24)(10,16,19,32)(12,23,14,21)(13,27,29,38)(17,33,30,31)(18,34,35,40)(20,36,37,26)(22,25,39,28);(2,3)(6,7)(9,12,15,14)(10,17,19,30)(11,21,24,23)(13,25,29,28)(16,31,32,33)(18,26,35,36)(20,34,37,40)(22,38,39,27)

Conjugacy classes & character table

3796 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(18688, 4664);