C₁₄₉×#7406 #49186355

order	38144
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₁₄₉ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₁₄₉×#7406

Subgroups

	subgroup	G/H
center	C₄×C₁₄₉	#353
commutator	C₂×M₁₆	C₂×C₄×C₁₄₉
Frattini, cosocle	#408	C₂²×C₁₄₉
socle	C₂×C₁₄₉	#1343
prime radical	#2017×C₁₄₉	C₂
LCS	G ▷ C₂×M₁₆ ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₁₄₉ ◁ C₂×C₄×C₁₄₉ ◁ C₂²×C₄×C₁₄₉ ◁ #408×C₁₄₉ ◁ G

244 conjugacy classes of subgroups; only the 42 normal subgroups are shown.

normal subgroups only

This group has 244 conjugacy classes of subgroups — too many to draw. Only its 42 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:

(1,2,5,10)(3,13,15,42)(4,9,20,33)(6,26,25,56)(7,11,24,39)(8,28,29,18)(12,40,38,19)(14,31,44,36)(16,49,48,64)(17,43,47,62)(21,35,53,37)(22,54,52,63)(23,55,27,32)(30,57,58,50)(34,60,59,51)(41,46,61,45)(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)(2,8)(4,18)(5,23)(6,27)(7,17)(9,32)(10,36)(11,30)(12,41)(13,34)(14,22)(15,40)(16,19)(20,46)(21,50)(24,48)(25,47)(26,31)(28,52)(29,38)(33,42)(35,49)(37,62)(39,61)(43,54)(44,53)(45,51)(55,63)(56,58)(57,60)(59,64)

Conjugacy classes & character table

4619 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(38144, 394);