C₁₄₉×#7425 #49186374

order	38144
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₁₄₉ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₁₄₉×#7425

Subgroups

	subgroup	G/H
center, socle	C₂×C₁₄₉	#1347
commutator	C₂×C₁₆	C₂×C₄×C₁₄₉
Frattini, cosocle	C₂×M₃₂	C₂²×C₁₄₉
prime radical	C₂³×C₁₄₉	SD₃₂
LCS	G ▷ C₂×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₄₉ ◁ C₂×C₄×C₁₄₉ ◁ C₂²×C₈×C₁₄₉ ◁ C₂×M₃₂×C₁₄₉ ◁ G

136 conjugacy classes of subgroups; only the 46 normal subgroups are shown.

normal subgroups only

This group has 136 conjugacy classes of subgroups — too many to draw. Only its 46 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) = 213
transitive degree	μ_t(G) = 9536

A minimal faithful permutation representation — 2 generators on 213 points:

(1,2,5,12,9,14,8,35)(3,15,17,49,21,50,20,56)(4,22,25,57,28,48,29,58)(6,31,30,47,7,33,24,13)(10,36,37,52,40,53,39,16)(11,41,43,26,45,60,44,27)(18,54,32,63,19,55,34,42)(23,46,59,62,51,64,61,38)(65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213);(1,3,9,21)(2,10,14,40)(4,23,28,51)(5,20,8,17)(6,32,7,34)(11,42,45,63)(12,37,35,39)(13,46,47,64)(15,48,50,22)(16,26,52,27)(18,24,19,30)(25,61,29,59)(31,62,33,38)(36,41,53,60)(43,54,44,55)(49,58,56,57)

Conjugacy classes & character table

4619 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(38144, 413);