C₁₄₉×#7260 #49186209

order	38144
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₄²×C₁₄₉ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₁₄₉×#7260

Subgroups

	subgroup	G/H
center	C₂×C₄×C₁₄₉	#102
commutator	C₂×C₈	C₄²×C₁₄₉
Frattini, cosocle	C₂³×C₈	C₂²×C₁₄₉
socle	C₂²×C₁₄₉	#340
prime radical	C₂⁴×C₁₄₉	SD₁₆
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₄₉ ◁ C₂³×C₄×C₁₄₉ ◁ C₂³×C₈×C₁₄₉ ◁ G

336 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 106 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) = 189
transitive degree	μ_t(G) = 9536

A minimal faithful permutation representation — 2 generators on 189 points:

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

Conjugacy classes & character table

8642 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(38144, 248);