C₁₃₇×#7065 #46856451

order	35072
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂×C₈×C₁₃₇ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₁₃₇×#7065

Subgroups

	subgroup	G/H
center	C₂×C₄×C₁₃₇	#99
commutator	C₂²×C₄	C₂×C₈×C₁₃₇
Frattini, cosocle	C₂×C₄⋊C₈	C₂²×C₁₃₇
socle	C₂²×C₁₃₇	#324
prime radical	C₂³×C₁₃₇	C₂²⋊C₈
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₃₇ ◁ C₂²×C₄×C₁₃₇ ◁ C₂×C₄⋊C₈×C₁₃₇ ◁ G

156 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 156 conjugacy classes of subgroups — too many to draw. Only its 54 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) = 8768

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

(1,2)(3,9,11,15)(4,12,13,7)(5,8)(6,10)(14,16)(17,18,20,25,21,26,31,39,22,24,30,36,32,37,43,46)(19,23,28,33,29,35,42,44,27,34,40,38,41,45,48,47)(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)(2,6,8,14)(4,10)(5,11)(7,9,12,15)(13,16)(17,19)(18,23,24,34)(20,28)(21,29)(22,27)(25,38,36,33)(26,35,37,45)(30,40)(31,42)(32,41)(39,47,46,44)(43,48)

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, 53);