C₁₃₇×#8307 #46857693

order	35072
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₁₃₇ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₁₃₇×#8307

Subgroups

	subgroup	G/H
center, socle	C₂³×C₁₃₇	C₂×C₂²⋊C₄
commutator	C₂⁴	C₂²×C₄×C₁₃₇
Frattini, cosocle	C₂³×C₄	C₂³×C₁₃₇
prime radical	C₂²×#121×C₁₃₇	C₂
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃₇ ◁ C₂⁴×C₄×C₁₃₇ ◁ G

1518 conjugacy classes of subgroups; only the 190 normal subgroups are shown.

normal subgroups only

This group has 1518 conjugacy classes of subgroups — too many to draw. Only its 190 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) = 157
transitive degree	μ_t(G) = 8768

A minimal faithful permutation representation — 3 generators on 157 points:

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

Conjugacy classes & character table

7124 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(35072, 1295);