C₁₄₉×#7385 #49186334

order	38144
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂×C₈×C₁₄₉ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₁₄₉×#7385

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₄₉	#357
commutator	C₁₆	C₂×C₈×C₁₄₉
Frattini, cosocle	C₄×C₁₆	C₂²×C₁₄₉
prime radical	C₂×D₁₆×C₁₄₉	C₄
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₄₉ ◁ C₄²×C₁₄₉ ◁ C₄×C₈×C₁₄₉ ◁ C₄×C₁₆×C₁₄₉ ◁ G

136 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 136 conjugacy classes of subgroups — too many to draw. Only its 58 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) = 19072

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

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

Conjugacy classes & character table

7748 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(38144, 373);