C₁₄₉×#11896 #49190845

order	38144
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₁₄₉ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₁₄₉×#11896

Subgroups

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

390 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 390 conjugacy classes of subgroups — too many to draw. Only its 118 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 — 3 generators on 189 points:

(1,2)(3,7,8,5)(4,6)(9,10,13,19,15,21,14,20)(11,22,24,38,26,32,25,36)(12,27,29,18,31,37,30,17)(16,28,33,23,35,39,34,40)(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);(1,3)(2,5)(4,8)(6,7)(9,11,15,26)(10,16,21,35)(12,28,31,39)(13,25,14,24)(17,36,18,38)(19,34,20,33)(22,27,32,37)(23,29,40,30);(3,8)(5,7)(9,12,13,29,15,31,14,30)(10,17,20,37,21,18,19,27)(11,23,24,39,26,40,25,28)(16,32,34,38,35,22,33,36)

Conjugacy classes & character table

5960 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(38144, 4884);