C₁₂₇×#13579 #44577252

order	32512
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₁₂₇ · Schur multiplier (H₂) = C₂²×C₄

Aliases

direct product

C₁₂₇×#13579

Subgroups

	subgroup	G/H
center, socle	C₂×C₁₂₇	C₂×#353
commutator	C₄²	C₂²×C₄×C₁₂₇
Frattini, cosocle	#124	C₂³×C₁₂₇
prime radical	C₁₂₇×#3151	C₂
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₂₇ ◁ C₂×C₄×C₁₂₇ ◁ C₂²×C₄×C₁₂₇ ◁ #532×C₁₂₇ ◁ G

454 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 454 conjugacy classes of subgroups — too many to draw. Only its 90 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) = 143
transitive degree	μ_t(G) = 2032

A minimal faithful permutation representation — 3 generators on 143 points:

(1,2,4,9)(3,7,11,14)(5,8,12,15)(6,10,13,16)(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);(2,7)(3,5)(4,12)(8,10)(11,13)(14,15);(1,3,6,5)(2,7,10,8)(4,11,13,12)(9,14,16,15)

Conjugacy classes & character table

3556 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(32512, 6567);