C₇₃×#13646 #32923050

order	18688
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂²×C₁₆×C₇₃ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₇₃×#13646

Subgroups

	subgroup	G/H
center	C₂×C₁₆×C₇₃	C₂³
commutator	C₂²	C₂²×C₁₆×C₇₃
Frattini, cosocle	C₂×C₁₆	C₂³×C₇₃
socle	C₂²×C₇₃	C₂²×C₁₆
prime radical	C₂×D₄×C₇₃	C₁₆
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₁₆×C₇₃ ◁ G

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

normal subgroups only

This group has 202 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)(5,6,9,14,10,15,23,30,11,7,16,18,24,19,31,33,8,12,20,25,21,26,34,35,22,17,13,27,28,32,29,36)(37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109);(2,4)(6,12)(7,17)(14,25)(15,26)(18,27)(19,32)(30,35)(33,36);(5,7,9,18,10,19,23,33,11,12,16,25,24,26,31,35,8,17,20,27,21,32,34,36,22,6,13,14,28,15,29,30)

Conjugacy classes & character table

8176 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(18688, 6634);