C₁₄₉×#9439 #49188388

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

Aliases

direct product

C₁₄₉×#9439

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₄₉	#386
commutator	C₄²	C₂²×C₄×C₁₄₉
Frattini, cosocle	C₂×C₄²	C₂³×C₁₄₉
prime radical	C₂²×D₄×C₁₄₉	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₄₉ ◁ C₂×C₄²×C₁₄₉ ◁ G

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

normal subgroups only

This group has 490 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) = 181
transitive degree	μ_t(G) = 4768

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

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

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, 2427);