#1609×C₁₆₇ #36319969

order	21376
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂³×C₁₆₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product

#1609×C₁₆₇

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₆₇	#120
commutator, Frattini, cosocle	C₂²×C₄	C₂³×C₁₆₇
prime radical	#121×C₁₆₇	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₆₇ ◁ C₂²×C₄×C₁₆₇ ◁ G

216 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 64 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) = 191
transitive degree	μ_t(G) = 5344

A minimal faithful permutation representation — 3 generators on 191 points:

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

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, 341);