#1625×C₁₆₇ #36319985

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

Aliases

direct product

#1625×C₁₆₇

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₆₇	#120
commutator, Frattini, cosocle	C₄²	C₂³×C₁₆₇
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₆₇ ◁ C₄²×C₁₆₇ ◁ G

256 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 72 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) = 183
transitive degree	μ_t(G) = 10688

A minimal faithful permutation representation — 3 generators on 183 points:

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

Conjugacy classes & character table

4342 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(21376, 357);