#2049×C₁₆₇ #36320409

order	21376
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂³×C₁₆₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

#2049×C₁₆₇

Subgroups

	subgroup	G/H
center, socle	C₂×C₁₆₇	#394
commutator, Frattini, cosocle	C₂²×C₄	C₂³×C₁₆₇
prime radical	C₂×#41×C₁₆₇	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₆₇ ◁ C₂²×C₄×C₁₆₇ ◁ G

254 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 254 conjugacy classes of subgroups — too many to draw. Only its 80 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) = 199
transitive degree	μ_t(G) = 5344

A minimal faithful permutation representation — 3 generators on 199 points:

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

Conjugacy classes & character table

3841 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(21376, 781);