C₁₄₉×#7346 #49186295

order	38144
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂×C₈×C₁₄₉ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₁₄₉×#7346

Subgroups

	subgroup	G/H
center	C₄×C₁₄₉	AΣL(1,𝔽₁₆)
commutator	C₂×D₄	C₂×C₈×C₁₄₉
Frattini, cosocle	#408	C₂²×C₁₄₉
socle	C₂×C₁₄₉	#1316
prime radical	#585×C₁₄₉	C₄
LCS	G ▷ C₂×D₄ ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₁₄₉ ◁ C₂×C₄×C₁₄₉ ◁ C₂²×C₄×C₁₄₉ ◁ #408×C₁₄₉ ◁ G

318 conjugacy classes of subgroups; only the 46 normal subgroups are shown.

normal subgroups only

This group has 318 conjugacy classes of subgroups — too many to draw. Only its 46 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) = 213
transitive degree	μ_t(G) = 9536

A minimal faithful permutation representation — 2 generators on 213 points:

(1,2,5,11,7,12,25,42,8,13,26,43,6,27,30,41)(3,14,16,44,17,45,23,57,18,46,52,60,28,53,51,64)(4,19,20,54,21,10,36,37,22,38,24,58,29,39,56,63)(9,31,32,50,34,59,61,48,35,55,62,15,33,47,40,49)(65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213);(1,3)(2,9)(4,15)(5,23)(6,28)(7,17)(8,18)(10,14)(11,40)(12,34)(13,35)(16,30)(19,53)(20,55)(21,49)(22,50)(24,31)(25,52)(26,51)(27,33)(29,48)(32,42)(36,47)(37,57)(38,45)(39,46)(41,62)(43,61)(44,54)(56,59)(58,60)(63,64)

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, 334);