C₇₃×#13280 #32922684

order	18688
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇₃ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₇₃×#13280

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇₃	C₂×#102
commutator	C₂×C₈	C₂²×C₄×C₇₃
Frattini, cosocle	C₂²×C₈	C₂³×C₇₃
prime radical	C₂³×C₇₃	C₂×SD₁₆
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇₃ ◁ C₂×C₄²×C₇₃ ◁ C₂×C₄×C₈×C₇₃ ◁ G

334 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 150 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)(3,7)(4,5)(6,8)(9,10,14,19,22,36,28,18,13,17,29,35,15,20,12,27)(11,21,25,16,31,33,23,38,24,39,40,32,26,37,30,34)(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);(2,5)(6,7)(9,11,22,26,13,24,15,31)(10,16,20,38,17,32,36,34)(12,25,14,30,28,40,29,23)(18,37,19,21,27,33,35,39);(1,3,4,8)(2,6,5,7)(10,17)(16,32)(18,27)(19,35)(20,36)(21,39)(33,37)(34,38)

Conjugacy classes & character table

3358 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(18688, 6268);