#1389×C₂₁₁ #41009009

order	27008
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₄²×C₂₁₁ · Schur multiplier (H₂) = C₂²

Aliases

direct product

#1389×C₂₁₁

Subgroups

	subgroup	G/H
center	C₄×C₂₁₁	#102
commutator	C₈	C₄²×C₂₁₁
Frattini, cosocle	C₂²×C₈	C₂²×C₂₁₁
socle	C₂×C₂₁₁	#340
prime radical	C₂³×C₂₁₁	SD₁₆
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₂₁₁ ◁ C₂²×C₄×C₂₁₁ ◁ C₂²×C₈×C₂₁₁ ◁ G

128 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 128 conjugacy classes of subgroups — too many to draw. Only its 76 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) = 243
transitive degree	μ_t(G) = 6752

A minimal faithful permutation representation — 2 generators on 243 points:

(1,2,5,10,7,11,6,22)(3,12,14,29,17,25,16,31)(4,18,19,27,21,32,20,9)(8,23,24,15,26,30,28,13)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243);(1,3,4,15,5,14,19,30,7,17,21,13,6,16,20,23)(2,8,9,25,22,28,32,29,11,26,27,12,10,24,18,31)

Conjugacy classes & character table

6752 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(27008, 121);