C₁₄₉×#10022 #49188971

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

Aliases

direct product

C₁₄₉×#10022

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₄₉	#392
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₁₄₉
prime radical	#2631×C₁₄₉	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₄₉ ◁ C₂×C₄²×C₁₄₉ ◁ G

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

normal subgroups only

This group has 576 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) = 181
transitive degree	μ_t(G) = 4768

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

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

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, 3010);