C₁₄₉×#13079 #49192028

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

Aliases

direct product

C₁₄₉×#13079

Subgroups

	subgroup	G/H
center, socle	C₂×C₁₄₉	#1893
commutator	C₂×C₈	C₂²×C₄×C₁₄₉
Frattini, cosocle	C₂²×C₈	C₂³×C₁₄₉
prime radical	#3058×C₁₄₉	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₄₉ ◁ C₂²×C₄×C₁₄₉ ◁ C₂×#131×C₁₄₉ ◁ G

486 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 486 conjugacy classes of subgroups — too many to draw. Only its 146 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:

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

Conjugacy classes & character table

6854 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(38144, 6067);