C₇₃×#11845 #32921249

order	18688
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇₃ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₇₃×#11845

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇₃	C₂×#99
commutator	C₂²×C₄	C₂²×C₄×C₇₃
Frattini, cosocle	C₂×C₂²⋊C₄	C₂³×C₇₃
prime radical	C₂⁵×C₇₃	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇₃ ◁ C₂⁴×C₇₃ ◁ C₂²×C₂²⋊C₄×C₇₃ ◁ G

506 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 506 conjugacy classes of subgroups — too many to draw. Only its 102 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) = 105
transitive degree	μ_t(G) = 2336

A minimal faithful permutation representation — 3 generators on 105 points:

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

Conjugacy classes & character table

2701 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(18688, 4833);