C₁₃₇×#10210 #46859596

order	35072
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂³×C₁₃₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₁₃₇×#10210

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₃₇	#392
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₁₃₇
prime radical	#532×C₁₃₇	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃₇ ◁ C₂×C₄²×C₁₃₇ ◁ G

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

normal subgroups only

This group has 440 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) = 185
transitive degree	μ_t(G) = 8768

A minimal faithful permutation representation — 3 generators on 185 points:

(3,10)(4,5)(7,11)(8,14)(12,13)(15,16)(17,18)(19,29)(20,34)(21,27)(22,26)(23,28)(24,31)(25,35)(30,41)(32,40)(33,39)(36,43)(37,42)(38,46)(44,45)(47,48)(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185);(1,2)(3,11)(4,8)(5,14)(6,9)(7,13)(10,15)(12,16)(17,19,23,33)(18,24,28,40)(20,30,36,46)(21,32,22,31)(25,41,42,38)(26,39,27,29)(34,44,43,48)(35,47,37,45);(1,3)(2,7)(4,13)(5,10)(6,12)(8,16)(9,15)(11,14)(17,20,23,36)(18,25,28,42)(19,30,33,46)(21,35,22,37)(24,38,40,41)(26,34,27,43)(29,44,39,48)(31,45,32,47)

Conjugacy classes & character table

3425 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(35072, 3198);