C₁₄₉×#10045 #49188994

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

Aliases

direct product

C₁₄₉×#10045

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₄₉	#392
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₁₄₉
prime radical	C₂³×C₁₄₉	#121
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₄₉ ◁ C₂×C₄²×C₁₄₉ ◁ G

444 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 98 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) = 197
transitive degree	μ_t(G) = 19072

A minimal faithful permutation representation — 3 generators on 197 points:

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

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, 3033);