C₁₃₇×#12201 #46861587

order	35072
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₁₃₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₁₃₇×#12201

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₃₇	B(4,𝔽₂)
commutator, Frattini, cosocle	C₂×C₄⋊C₄	C₂³×C₁₃₇
prime radical	#537×C₁₃₇	C₂²
LCS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃₇ ◁ C₂³×C₁₃₇ ◁ C₂×C₄⋊C₄×C₁₃₇ ◁ G

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

normal subgroups only

This group has 380 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) = 177
transitive degree	μ_t(G) = 8768

A minimal faithful permutation representation — 3 generators on 177 points:

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

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, 5189);