C₇₃×#11443 #32920847

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

Aliases

direct product

C₇₃×#11443

Subgroups

	subgroup	G/H
center	C₂×C₄×C₇₃	C₂×C₂²⋊C₄
commutator	C₂×C₄	C₂²×C₈×C₇₃
Frattini, cosocle	C₂²×C₈	C₂³×C₇₃
socle	C₂²×C₇₃	C₂×C₂²⋊C₈
prime radical	C₂⁴×C₇₃	M₁₆
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₇₃ ◁ C₂³×C₈×C₇₃ ◁ G

382 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 382 conjugacy classes of subgroups — too many to draw. Only its 142 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 — 3 generators on 113 points:

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

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, 4431);