C₇₃×#13634 #32923038

order	18688
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇₃ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₇₃×#13634

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇₃	C₂×D₁₆
commutator	C₁₆	C₂²×C₄×C₇₃
Frattini, cosocle	C₂×C₁₆	C₂³×C₇₃
prime radical	C₂³×C₇₃	D₁₆
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇₃ ◁ C₂²×C₄×C₇₃ ◁ C₂²×C₈×C₇₃ ◁ C₂²×C₁₆×C₇₃ ◁ G

194 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 194 conjugacy classes of subgroups — too many to draw. Only its 126 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) = 109
transitive degree	μ_t(G) = 9344

A minimal faithful permutation representation — 3 generators on 109 points:

(1,2,3,4)(6,11)(7,12)(8,16)(9,19)(13,21)(14,30)(15,31)(17,33)(18,27)(20,32)(22,23)(24,29)(25,26)(28,36)(34,35)(37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109);(5,6,12,23,8,14,25,31,19,29,33,36,32,34,18,21,10,13,27,35,20,28,17,24,9,15,26,30,16,22,7,11);(6,13)(11,21)(14,28)(15,29)(22,34)(23,35)(24,31)(30,36)

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, 6622);