C₁₄₉×#13630 #49192579

order	38144
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂²×C₁₆×C₁₄₉ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₁₄₉×#13630

Subgroups

	subgroup	G/H
center	C₁₆×C₁₄₉	C₂×D₄
commutator	C₄	C₂²×C₁₆×C₁₄₉
Frattini, cosocle	C₂×C₁₆	C₂³×C₁₄₉
socle	C₂×C₁₄₉	C₂×C₄⋊C₁₆
prime radical	C₂³×C₁₄₉	M₃₂
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₁₆×C₁₄₉ ◁ C₂²×C₁₆×C₁₄₉ ◁ G

198 conjugacy classes of subgroups; only the 154 normal subgroups are shown.

normal subgroups only

This group has 198 conjugacy classes of subgroups — too many to draw. Only its 154 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 — 3 generators on 213 points:

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

Conjugacy classes & character table

13112 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(38144, 6618);