C₇₃×#7086 #32916490

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

Aliases

direct product

C₇₃×#7086

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇₃	#327
commutator	C₂²×C₈	C₂×C₄×C₇₃
Frattini, cosocle	C₂×C₂²⋊C₈	C₂²×C₇₃
prime radical	C₂⁴×C₇₃	SD₁₆
LCS	G ▷ C₂²×C₈ ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₇₃ ◁ C₂²×C₄×C₇₃ ◁ C₂×C₂²⋊C₈×C₇₃ ◁ G

264 conjugacy classes of subgroups; only the 50 normal subgroups are shown.

normal subgroups only

This group has 264 conjugacy classes of subgroups — too many to draw. Only its 50 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) = 4672

A minimal faithful permutation representation — 2 generators on 113 points:

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

Conjugacy classes & character table

2482 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(18688, 74);