C₁₃₇×#7220 #46856606

order	35072
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₄²×C₁₃₇ · Schur multiplier (H₂) = C₄

Aliases

direct product

C₁₃₇×#7220

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₃₇	#340
commutator	C₂×C₈	C₄²×C₁₃₇
Frattini, cosocle	C₂×C₄×C₈	C₂²×C₁₃₇
prime radical	C₂³×C₁₃₇	#103
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃₇ ◁ C₂×C₄²×C₁₃₇ ◁ C₂×C₄×C₈×C₁₃₇ ◁ G

188 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 188 conjugacy classes of subgroups — too many to draw. Only its 90 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) = 185
transitive degree	μ_t(G) = 17536

A minimal faithful permutation representation — 2 generators on 185 points:

(1,2)(3,9,11,15)(4,7)(5,8)(6,10,13,16)(12,14)(17,18,21,27,24,29,23,39)(19,30,32,45,34,46,37,41)(20,28,22,38,36,44,35,26)(25,33,40,31,43,47,42,48)(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185);(1,3)(2,6,8,13)(4,10)(5,11)(7,9,14,15)(12,16)(17,19,22,33,21,37,36,48,24,34,35,47,23,32,20,31)(18,25,28,41,39,40,26,30,29,43,44,45,27,42,38,46)

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, 208);