C₁₄₉×#7399 #49186348

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

Aliases

direct product

C₁₄₉×#7399

Subgroups

	subgroup	G/H
center, socle	C₂×C₁₄₉	#1343
commutator	C₂²×C₈	C₂×C₄×C₁₄₉
Frattini, cosocle	#408	C₂²×C₁₄₉
prime radical	C₂×#41×C₁₄₉	D₄
LCS	G ▷ C₂²×C₈ ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₄₉ ◁ C₂×C₄×C₁₄₉ ◁ C₂²×C₄×C₁₄₉ ◁ #408×C₁₄₉ ◁ G

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

normal subgroups only

This group has 238 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)(3,13,15,43)(4,18,21,38)(6,12,26,39)(7,11,25,40)(8,28,29,52)(9,32,35,23)(14,31,46,37)(16,33,49,61)(17,44,48,58)(19,30,51,59)(20,34,53,57)(22,42,54,64)(24,55,27,50)(36,56,62,45)(41,60,63,47)(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);(1,3,7,17)(2,8,11,30)(4,19,23,52)(5,24,25,49)(6,27,20,16)(9,33,18,50)(10,37,40,63)(12,41,34,31)(13,42,44,36)(14,45,47,22)(15,26,48,53)(21,46,32,60)(28,56,51,54)(29,57,59,39)(35,58,38,43)(55,62,61,64)

Conjugacy classes & character table

3725 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(38144, 387);