C₁₄₉×#13566 #49192515

order	38144
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₁₄₉ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₁₄₉×#13566

Subgroups

	subgroup	G/H
center, socle	C₂×C₁₄₉	C₂×AΣL(1,𝔽₁₆)
commutator	C₂×D₄	C₂²×C₄×C₁₄₉
Frattini, cosocle	#124	C₂³×C₁₄₉
prime radical	#3018×C₁₄₉	C₂
LCS	G ▷ C₂×D₄ ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₄₉ ◁ C₂×C₄×C₁₄₉ ◁ C₂²×C₄×C₁₄₉ ◁ #532×C₁₄₉ ◁ G

478 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 478 conjugacy classes of subgroups — too many to draw. Only its 90 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:

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

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, 6554);