C₁₃₇×#12872 #46862258

order	35072
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₁₃₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₁₃₇×#12872

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₃₇	C₂×#102
commutator	C₂×C₈	C₂²×C₄×C₁₃₇
Frattini, cosocle	C₂²×C₈	C₂³×C₁₃₇
prime radical	C₂⁴×C₁₃₇	SD₁₆
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃₇ ◁ C₂³×C₄×C₁₃₇ ◁ C₂³×C₈×C₁₃₇ ◁ G

398 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 398 conjugacy classes of subgroups — too many to draw. Only its 142 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:

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

Conjugacy classes & character table

6302 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(35072, 5860);