C₂×#1425×C₁₄₉ #49192584

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

Aliases

direct product

C₂×#1425×C₁₄₉

Subgroups

	subgroup	G/H
center	C₂×C₄×C₁₄₉	D₁₆
commutator	C₁₆	C₂²×C₄×C₁₄₉
Frattini, cosocle	C₂×C₁₆	C₂³×C₁₄₉
socle	C₂²×C₁₄₉	#366
prime radical	C₂³×C₁₄₉	Q₃₂
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₄₉ ◁ C₂²×C₄×C₁₄₉ ◁ C₂²×C₈×C₁₄₉ ◁ C₂²×C₁₆×C₁₄₉ ◁ G

182 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 182 conjugacy classes of subgroups — too many to draw. Only its 126 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) = 215
transitive degree	μ_t(G) = 19072

A minimal faithful permutation representation — 3 generators on 215 points:

(1,2)(67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215);(3,4,7,13,10,15,9,31)(5,16,19,41,22,43,21,53)(6,23,25,54,27,56,26,57)(8,29,28,39,30,58,24,14)(11,32,33,49,35,48,59,18)(12,36,37,46,38,45,55,17)(20,51,50,63,52,66,47,42)(34,44,40,60,61,65,62,64);(3,5,17,44,8,20,25,32,7,19,36,60,28,50,27,49,10,22,46,65,30,52,26,48,9,21,45,64,24,47,6,18)(4,11,23,42,14,34,55,53,31,59,57,66,58,62,38,43,15,35,56,63,39,61,37,41,13,33,54,51,29,40,12,16)

Conjugacy classes & character table

11324 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(38144, 6623);