C₁₃₇×#12109 #46861495

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

Aliases

direct product

C₁₃₇×#12109

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₃₇	#447
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₁₃₇
prime radical	#2018×C₁₃₇	C₂
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃₇ ◁ C₂²×C₄×C₁₃₇ ◁ #408×C₁₃₇ ◁ G

410 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 410 conjugacy classes of subgroups — too many to draw. Only its 74 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:

(5,8)(10,16)(11,20)(12,23)(13,14)(17,31)(18,35)(19,30)(21,37)(22,27)(24,28)(25,26)(29,32)(33,40)(34,38)(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)(3,5)(4,6)(7,8)(9,10,15,19)(11,17,22,34)(12,24,26,35)(13,18,14,28)(16,25,30,23)(20,32,27,33)(21,38,37,31)(29,39,40,36);(1,3)(2,5)(4,7)(6,8)(9,11)(10,17)(12,21)(13,27)(14,20)(15,22)(16,29)(18,33)(19,34)(23,36)(24,38)(25,39)(26,37)(28,32)(30,40)(31,35)

Conjugacy classes & character table

4247 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(35072, 5097);