C₁₄₉×#9007 #49187956

order	38144
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₁₄₉ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₁₄₉×#9007

Subgroups

	subgroup	G/H
center	C₂×C₄×C₁₄₉	#120
commutator	C₄²	C₂²×C₄×C₁₄₉
Frattini, cosocle	C₂×C₄²	C₂³×C₁₄₉
socle	C₂²×C₁₄₉	#386
prime radical	#3150×C₁₄₉	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₄₉ ◁ C₂×C₄²×C₁₄₉ ◁ G

502 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 502 conjugacy classes of subgroups — too many to draw. Only its 118 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) = 189
transitive degree	μ_t(G) = 19072

A minimal faithful permutation representation — 3 generators on 189 points:

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

Conjugacy classes & character table

5960 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(38144, 1995);