C₁₄₉×#7400 #49186349

order	38144
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₁₄₉ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₁₄₉×#7400

Subgroups

	subgroup	G/H
center, socle	C₂×C₁₄₉	#1343
commutator	C₂²×C₈	C₂×C₄×C₁₄₉
Frattini, cosocle	#408	C₂²×C₁₄₉
prime radical	#2018×C₁₄₉	C₂
LCS	G ▷ C₂²×C₈ ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₄₉ ◁ C₂×C₄×C₁₄₉ ◁ C₂²×C₄×C₁₄₉ ◁ #408×C₁₄₉ ◁ G

254 conjugacy classes of subgroups; only the 42 normal subgroups are shown.

normal subgroups only

This group has 254 conjugacy classes of subgroups — too many to draw. Only its 42 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 — 2 generators on 181 points:

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

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, 388);