C₁₄₉×#13354 #49192303

order	38144
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₁₄₉ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₁₄₉×#13354

Subgroups

	subgroup	G/H
center, socle	C₂×C₁₄₉	#2021
commutator, Frattini, cosocle	C₂×#41	C₂³×C₁₄₉
prime radical	#3021×C₁₄₉	C₂
LCS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₄₉ ◁ C₂×C₄×C₁₄₉ ◁ C₂×#41×C₁₄₉ ◁ G

692 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 692 conjugacy classes of subgroups — too many to draw. Only its 94 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:

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

Conjugacy classes & character table

4172 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(38144, 6342);