C₁₄₉×#10525 #49189474

order	38144
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂³×C₁₄₉ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₁₄₉×#10525

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₄₉	#394
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₁₄₉
prime radical	#3253×C₁₄₉	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₄₉ ◁ C₂×C₄²×C₁₄₉ ◁ G

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

normal subgroups only

This group has 370 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) = 181
transitive degree	μ_t(G) = 19072

A minimal faithful permutation representation — 3 generators on 181 points:

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

Conjugacy classes & character table

4172 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(38144, 3513);