C₁₄₉×#9070 #49188019

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

Aliases

direct product

C₁₄₉×#9070

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	#532×C₁₄₉	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₄₉ ◁ C₂×C₄²×C₁₄₉ ◁ G

426 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 426 conjugacy classes of subgroups — too many to draw. Only its 126 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:

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

Conjugacy classes & character table

7748 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(38144, 2058);