C₇₃×#13356 #32922760

order	18688
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇₃ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₇₃×#13356

Subgroups

	subgroup	G/H
center, socle	C₂×C₇₃	#2021
commutator, Frattini, cosocle	C₂×#41	C₂³×C₇₃
prime radical	#585×C₇₃	C₂²
LCS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₇₃ ◁ C₂×C₄×C₇₃ ◁ C₂×#41×C₇₃ ◁ G

676 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 676 conjugacy classes of subgroups — too many to draw. Only its 94 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) = 137
transitive degree	μ_t(G) = 4672

A minimal faithful permutation representation — 3 generators on 137 points:

(1,2,7,12)(3,13,17,35)(4,18,21,37)(5,22,25,10)(6,26,28,11)(8,23,31,15)(9,27,34,20)(14,40,43,64)(16,46,48,39)(19,51,53,50)(24,36,58,55)(29,60,32,63)(30,45,59,56)(33,52,54,57)(38,47,61,42)(41,49,44,62)(65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137);(1,3)(2,8)(4,14)(5,23)(6,16)(7,17)(9,32)(10,35)(11,30)(12,31)(13,22)(15,25)(18,49)(19,44)(20,42)(21,43)(24,56)(26,59)(27,47)(28,48)(29,34)(33,61)(36,46)(37,62)(38,54)(39,55)(40,51)(41,53)(45,58)(50,64)(52,63)(57,60);(1,4,7,21)(2,9,12,34)(3,14,17,43)(5,19,25,53)(6,27,28,20)(8,29,31,32)(10,33,22,54)(11,37,26,18)(13,38,35,61)(15,44,23,41)(16,47,48,42)(24,57,58,52)(30,49,59,62)(36,51,55,50)(39,64,46,40)(45,60,56,63)

Conjugacy classes & character table

2044 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(18688, 6344);