C₁₄₉×#13609 #49192558

order	38144
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₁₄₉ · Schur multiplier (H₂) = C₂²×C₄

Aliases

direct product

C₁₄₉×#13609

Subgroups

	subgroup	G/H
center, socle	C₂×C₁₄₉	#2127
commutator	C₄²	C₂²×C₄×C₁₄₉
Frattini, cosocle	#127	C₂³×C₁₄₉
prime radical	#3220×C₁₄₉	C₂
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₄₉ ◁ C₂²×C₁₄₉ ◁ C₂²×C₄×C₁₄₉ ◁ #535×C₁₄₉ ◁ G

348 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 86 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) = 4768

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

(1,2)(3,12,14,7)(4,8)(5,10,20,9)(6,11)(13,27,28,21)(15,22,19,23)(16,29,30,32)(17,25)(18,24,31,26)(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);(1,3)(2,7)(4,16)(5,19)(6,15)(8,21)(9,22)(10,12)(11,23)(13,17)(14,20)(18,28)(24,27)(25,32)(26,29)(30,31);(1,4)(2,8)(3,13)(5,18)(6,17)(7,21)(9,26)(10,24)(11,25)(12,27)(14,28)(15,16)(19,30)(20,31)(22,29)(23,32)

Conjugacy classes & character table

3725 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(38144, 6597);