C₁₃₇×#7359 #46856745

order	35072
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂×C₈×C₁₃₇ · Schur multiplier (H₂) = C₄

Aliases

direct product

C₁₃₇×#7359

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₃₇	#354
commutator	C₄²	C₂×C₈×C₁₃₇
Frattini, cosocle	#382	C₂²×C₁₃₇
prime radical	C₂³×C₁₃₇	#99
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃₇ ◁ C₂³×C₁₃₇ ◁ C₂²×C₄×C₁₃₇ ◁ #382×C₁₃₇ ◁ G

176 conjugacy classes of subgroups; only the 50 normal subgroups are shown.

normal subgroups only

This group has 176 conjugacy classes of subgroups — too many to draw. Only its 50 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 — 2 generators on 177 points:

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

Conjugacy classes & character table

4658 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(35072, 347);