C₁₄₉×#13052 #49192001

order	38144
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₁₄₉ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₁₄₉×#13052

Subgroups

	subgroup	G/H
center	C₄×C₁₄₉	B(4,𝔽₂)
commutator	C₂×D₄	C₂²×C₄×C₁₄₉
Frattini, cosocle	C₂×#41	C₂³×C₁₄₉
socle	C₂×C₁₄₉	#1889
prime radical	#3018×C₁₄₉	C₂
LCS	G ▷ C₂×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₁₄₉ ◁ C₂×C₄×C₁₄₉ ◁ C₂×#41×C₁₄₉ ◁ G

624 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 624 conjugacy classes of subgroups — too many to draw. Only its 114 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:

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

Conjugacy classes & character table

5066 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(38144, 6040);