C₇₃×#12118 #32921522

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

Aliases

direct product

C₇₃×#12118

Subgroups

	subgroup	G/H
center, socle	C₂×C₇₃	#1595
commutator, Frattini, cosocle	C₂×M₁₆	C₂³×C₇₃
prime radical	C₇₃×#2017	C₂
LCS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₇₃ ◁ C₂×C₄×C₇₃ ◁ C₂×M₁₆×C₇₃ ◁ G

392 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 392 conjugacy classes of subgroups — too many to draw. Only its 70 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)(3,15)(4,20)(5,24)(6,13)(7,12)(8,14)(9,17)(10,22)(11,27)(16,44)(18,52)(19,43)(21,55)(23,40)(25,33)(26,36)(28,41)(29,37)(30,39)(31,38)(32,59)(34,50)(35,51)(42,46)(45,53)(47,64)(48,62)(49,60)(54,58)(56,61)(57,63)(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);(1,3,8,19)(2,9,14,35)(4,16,23,48)(5,25,28,50)(6,18,7,31)(10,32,37,61)(11,38,39,52)(12,34,13,33)(15,41,43,24)(17,30,51,27)(20,53,40,60)(21,56,26,59)(22,46,29,47)(36,62,55,44)(42,63,64,54)(45,57,49,58);(1,4)(2,10)(3,16)(5,26)(6,29)(7,22)(8,23)(9,32)(11,36)(12,40)(13,20)(14,37)(15,42)(17,49)(18,47)(19,48)(21,28)(24,54)(25,59)(27,57)(30,58)(31,46)(33,53)(34,60)(35,61)(38,62)(39,55)(41,63)(43,64)(44,52)(45,51)(50,56)

Conjugacy classes & character table

1825 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(18688, 5106);